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**Unformatted text preview: **11. Round off the following numbers to three significant figures: (a) 4.65735 m, (b) 55.578 s, (c) 4555 N, and (d) 2768 kg.
12. Represent each of the following combinations of units in the correct SI form using an appropriate prefix: (a) mMN, (b) N>mm, (c) MN>ks2, and (d) kN>ms.
13. Represent each of the following quantities in the correct SI form using an appropriate prefix: (a) 0.000431 kg, (b) 35.3(103) N, and (c) 0.00532 km.
*14. Represent each of the following combinations of units in the correct SI form: (a) Mg>ms, (b) N>mm, and (c) mN>(kg # ms).
1
15. Represent each of the following combinations of units in the correct SI form using an appropriate prefix: (a) kN>ms, (b) Mg>mN, and (c) MN>(kg # ms).
16. Represent each of the following to three significant figures and express each answer in SI units using an appropriate prefix: (a) 45 320 kN, (b) 568(105) mm, and (c) 0.005 63 mg.
17. A rocket has a mass of 250(103) slugs on earth. Specify (a) its mass in SI units and (b) its weight in SI units. If the rocket is on the moon, where the acceleration due to gravity is gm = 5.30 ft>s2, determine to three significant figures (c) its weight in SI units and (d) its mass in SI units.
FPO
*18. If a car is traveling at 55 mi>h, determine its speed in kilometers per hour and meters per second.
FPO
2
19. The pascal (Pa) is actually a very small unit of pressure. To show this, convert 1 Pa = 1 N>m2 to lb>ft2. Atmospheric pressure at sea level is 14.7 lb> in2. How many pascals is this?
FPO
110. What is the weight in newtons of an object that has a mass of: (a) 10 kg, (b) 0.5 g, and (c) 4.50 Mg? Express the result to three significant figures. Use an appropriate prefix.
111. Evaluate each of the following to three significant figures and express each answer in Sl units using an appropriate prefix: (a) 354 mg(45 km) > (0.0356 kN), (b) (0.004 53 Mg)(201 ms), and (c) 435 MN> 23.2 mm.
*112. The specific weight (wt.> vol.) of brass is 520 lb>ft3. Determine its density (mass> vol.) in SI units. Use an appropriate prefix.
3
113. Convert each of the following to three significant figures: (a) 20 lb # ft to N # m, (b) 450 lb>ft3 to kN>m3, and (c) 15 ft> h to mm> s.
114. The density (mass> volume) of aluminum is 5.26 slug>ft3. Determine its density in SI units. Use an appropriate prefix.
115. Water has a density of 1.94 slug>ft3. What is the density expressed in SI units? Express the answer to three significant figures.
*116. Two particles have a mass of 8 kg and 12 kg, respectively. If they are 800 mm apart, determine the force of gravity acting between them. Compare this result with the weight of each particle.
4
117. Determine the mass in kilograms of an object that has a weight of (a) 20 mN, (b) 150 kN, and (c) 60 MN. Express the answer to three significant figures.
118. Evaluate each of the following to three significant figures and express each answer in SI units using an appropriate prefix: (a) (200 kN)2, (b) (0.005 mm)2, and (c) (400 m)3.
119. Using the base units of the SI system, show that Eq. 12 is a dimensionally homogeneous equation which gives F in newtons. Determine to three significant figures the gravitational force acting between two spheres that are touching each other. The mass of each sphere is 200 kg and the radius is 300 mm.
5
*120. Evaluate each of the following to three significant figures and express each answer in SI units using an appropriate prefix: (a) (0.631 Mm)>(8.60 kg)2, and (b) (35 mm)2(48 kg)3.
121. Evaluate (204 mm)(0.00457 kg)> (34.6 N) to three significant figures and express the answer in SI units using an appropriate prefix.
6
21. If u = 30 and T = 6 kN, determine the magnitude of the resultant force acting on the eyebolt and its direction measured clockwise from the positive x axis.
y u
T
x 45
8 kN
7
22. If u = 60 and T = 5 kN, determine the magnitude of the resultant force acting on the eyebolt and its direction measured clockwise from the positive x axis.
y u
T
x 45
8 kN
8
23. If the magnitude of the resultant force is to be 9 kN directed along the positive x axis, determine the magnitude of force T acting on the eyebolt and its angle u.
y u
T
x 45
8 kN
9
*24. Determine the magnitude of the resultant force acting on the bracket and its direction measured counterclockwise from the positive u axis.
u
v 30 30 45 F1
F2
150 lb
200 lb
10
25. Resolve F1 into components along the u and v axes, and determine the magnitudes of these components.
v 30 u 30 45 F1
F2
150 lb
200 lb
11
26. Resolve F2 into components along the u and v axes, and determine the magnitudes of these components.
v 30 u 30 45 F1
F2
150 lb
200 lb
12
27. If FB = 2 kN and the resultant force acts along the positive u axis, determine the magnitude of the resultant force and the angle u.
y
FA u 30 A
3 kN x
B FB
u
13
*28. If the resultant force is required to act along the positive u axis and have a magnitude of 5 kN, determine the required magnitude of FB and its direction u.
y
FA u 30 A
3 kN x
B FB
u
14
29. The plate is subjected to the two forces at A and B as shown. If u = 60, determine the magnitude of the resultant of these two forces and its direction measured clockwise from the horizontal.
FA u
8 kN
A
40 B FB 6 kN
15
210. Determine the angle of u for connecting member A to the plate so that the resultant force of FA and FB is directed horizontally to the right.Also, what is the magnitude of the resultant force?
FA u
8 kN
A
40 B FB 6 kN
211. If the tension in the cable is 400 N, determine the magnitude and direction of the resultant force acting on the pulley. This angle is the same angle u of line AB on the tailboard block.
400 N 30
y
u B A x 400 N
16
*212. The device is used for surgical replacement of the knee joint. If the force acting along the leg is 360 N, determine its components along the x and y axes.
y
y 10
x x 60 360 N
17
213. The device is used for surgical replacement of the knee joint. If the force acting along the leg is 360 N, determine its components along the x and y axes.
y
y 10
x x 60 360 N
18
214. Determine the design angle u (0 ... u ... 90) for strut AB so that the 400-lb horizontal force has a component of 500 lb directed from A towards C. What is the component of force acting along member AB? Take f = 40.
400 lb A u f B
C
215. Determine the design angle f (0 ... f ... 90) between struts AB and AC so that the 400-lb horizontal force has a component of 600 lb which acts up to the left, in the same direction as from B towards A. Take u = 30.
400 lb A u f B
C
19
*216. Resolve F1 into components along the u and v axes and determine the magnitudes of these components.
v F1 250 N
F2
150 N 30 105
30
u
217. Resolve F2 into components along the u and v axes and determine the magnitudes of these components.
v F1 250 N
F2
150 N 30 105
30
u
20
218. The truck is to be towed using two ropes. Determine the magnitudes of forces FA and FB acting on each rope in order to develop a resultant force of 950 N directed along the positive x axis. Set u = 50.
y
A
FA 20 u FB x
B
219. The truck is to be towed using two ropes. If the resultant force is to be 950 N, directed along the positive x axis, determine the magnitudes of forces FA and FB acting on each rope and the angle u of FB so that the magnitude of FB is a minimum. FA acts at 20 from the x axis as shown.
y
A
FA 20 u FB x
B
21
*220. If f = 45, F1 = 5 kN, and the resultant force is 6 kN directed along the positive y axis, determine the required magnitude of F2 and its direction u.
F1 y f u F2
x
60
22
221. If f = 30 and the resultant force is to be 6 kN directed along the positive y axis, determine the magnitudes of F1 and F2 and the angle u if F2 is required to be a minimum.
F1 y f u F2
x
60
23
222. If f = 30, F1 = 5 kN, and the resultant force is to be directed along the positive y axis, determine the magnitude of the resultant force if F2 is to be a minimum. Also, what is F2 and the angle u?
F1 y f u F2
x
60
24
223. If u = 30 and F2 = 6 kN, determine the magnitude of the resultant force acting on the plate and its direction measured clockwise from the positive x axis.
y F3 5 kN F2
u F1 4 kN x
25
*224. If the resultant force FR is directed along a line measured 75 clockwise from the positive x axis and the magnitude of F2 is to be a minimum, determine the magnitudes of FR and F2 and the angle u ... 90.
y F3 5 kN F2
u F1 4 kN x
26
225. Two forces F1 and F2 act on the screw eye. If their lines of action are at an angle u apart and the magnitude of each force is F1 = F2 = F, determine the magnitude of the resultant force FR and the angle between FR and F1.
u
F1
F2
27
226. The log is being towed by two tractors A and B. Determine the magnitudes of the two towing forces FA and FB if it is required that the resultant force have a magnitude FR = 10 kN and be directed along the x axis. Set u = 15.
y
FA 30 u FB A x
B
227. The resultant FR of the two forces acting on the log is to be directed along the positive x axis and have a magnitude of 10 kN, determine the angle u of the cable, attached to B such that the magnitude of force FB in this cable is a minimum. What is the magnitude of the force in each cable for this situation?
y
FA 30 u FB A x
B
28
*228. The beam is to be hoisted using two chains. Determine the magnitudes of forces FA and FB acting on each chain in order to develop a resultant force of 600 N directed along the positive y axis. Set u = 45.
y FB u 30 FA
x
29
229. The beam is to be hoisted using two chains. If the resultant force is to be 600 N directed along the positive y axis, determine the magnitudes of forces FA and FB acting on each chain and the angle u of FB so that the magnitude of FB is a minimum. FA acts at 30 from the y axis, as shown.
y FB u 30 FA
x
30
230. Three chains act on the bracket such that they create a resultant force having a magnitude of 500 lb. If two of the chains are subjected to known forces, as shown, determine the angle u of the third chain measured clockwise from the positive x axis, so that the magnitude of force F in this chain is a minimum. All forces lie in the xy plane. What is the magnitude of F? Hint: First find the resultant of the two known forces. Force F acts in this direction.
y
300 lb
30 x u F
200 lb
31
231. Three cables pull on the pipe such that they create a resultant force having a magnitude of 900 lb. If two of the cables are subjected to known forces, as shown in the figure, determine the angle u of the third cable so that the magnitude of force F in this cable is a minimum. All forces lie in the xy plane. What is the magnitude of F? Hint: First find the resultant of the two known forces.
y 600 lb
45 F
u x 30
400 lb
32
*232. Determine the magnitude of the resultant force acting on the pin and its direction measured clockwise from the positive x axis.
y
F1
30 lb
45 15 F2 15 F3 25 lb
x 40 lb
33
233. If F1 = 600 N and f = 30, determine the magnitude of the resultant force acting on the eyebolt and its direction measured clockwise from the positive x axis.
y
F1 f x 60
5 3
4
F2 450 N
500 N
F3
34
234. If the magnitude of the resultant force acting on the eyebolt is 600 N and its direction measured clockwise from the positive x axis is u = 30, determine the magnitude of F and the angle f. 1
y
F1 f x 60
5 3
4
F2 450 N
500 N
F3
35
235. The contact point between the femur and tibia bones of the leg is at A. If a vertical force of 175 lb is applied at this point, determine the components along the x and y axes. Note that the y component represents the normal force on the load-bearing region of the bones. Both the x and y components of this force cause synovial fluid to be squeezed out of the bearing space.
y 175 lb
A
12 13 5
x
36
*236. If f = 30 and F2 = 3 kN, determine the magnitude of the resultant force acting on the plate and its direction u measured clockwise from the positive x axis.
y 30
F1
4 kN F2
f x
3 4
5
F3
5 kN
37
237. If the magnitude for the resultant force acting on the plate is required to be 6 kN and its direction measured clockwise from the positive x axis is u = 30, determine the magnitude of F2 and its direction f.
y 30
F1
4 kN F2
f x
3 4
5
F3
5 kN
38
238. If f = 30 and the resultant force acting on the gusset plate is directed along the positive x axis, determine the magnitudes of F2 and the resultant force.
y 30
F1
4 kN F2
f x
3 4
5
F3
5 kN
39
239. Determine the magnitude of F1 and its direction u so that the resultant force is directed vertically upward and has a magnitude of 800 N.
y 600 N
3 5 4
u
F1 400 N
30
x A
40
*240. Determine the magnitude and direction measured counterclockwise from the positive x axis of the resultant force of the three forces acting on the ring A. Take F1 = 500 N and u = 20.
y 600 N
3 5 4
u
F1 400 N
30
x A
41
241. Determine the magnitude and direction u of FB so that the resultant force is directed along the positive y axis and has a magnitude of 1500 N.
y
FB 30
FA
700 N
B
u
A
x
42
242. Determine the magnitude and angle measured counterclockwise from the positive y axis of the resultant force acting on the bracket if FB = 600 N and u = 20.
y
FB 30
FA
700 N
B
u
A
x
43
243. If f = 30 and F1 = 250 lb, determine the magnitude of the resultant force acting on the bracket and its direction measured clockwise from the positive x axis.
y F1
f x
3 4 13 12 5
5
F2
300 lb
F3
260 lb
44
*244. If the magnitude of the resultant force acting on the bracket is 400 lb directed along the positive x axis, determine the magnitude of F1 and its direction f.
y F1
f x
3 4 13 12 5
5
F2
300 lb
F3
260 lb
45
245. If the resultant force acting on the bracket is to be directed along the positive x axis and the magnitude of F1 is required to be a minimum, determine the magnitudes of the resultant force and F1.
y F1
f x
3 4 13 12 5
5
F2
300 lb
F3
260 lb
46
246. The three concurrent forces acting on the screw eye produce a resultant force FR = 0. If F2 = 2 F1 and F1 is to 3 be 90 from F2 as shown, determine the required magnitude of F3 expressed in terms of F1 and the angle u.
y F1
60 x 30
u
F2
F3
247. Determine the magnitude of FA and its direction u so that the resultant force is directed along the positive x axis and has a magnitude of 1250 N.
y
FA A u x O 30 B FB 800 N
47
*248. Determine the magnitude and direction measured counterclockwise from the positive x axis of the resultant force acting on the ring at O if FA = 750 N and u = 45.
y
FA A u x O 30 B FB 800 N
249. Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis.
y F1 = 60 lb
1 1 2
x 60 F2 70 lb F3 50 lb 45
48
250. The three forces are applied to the bracket. Determine the range of values for the magnitude of force P so that the resultant of the three forces does not exceed 2400 N.
800 N 3000 N 90 60 P
49
251. If F1 = 150 N and f = 30, determine the magnitude of the resultant force acting on the bracket and its direction measured clockwise from the positive x axis.
y F1 u
f
30 x F2
12 5 13
200 N
F3
260 N
50
*252. If the magnitude of the resultant force acting on the bracket is to be 450 N directed along the positive u axis, determine the magnitude of F1 and its direction f.
y F1 u
f
30 x F2
12 5 13
200 N
F3
260 N
51
253. If the resultant force acting on the bracket is required to be a minimum, determine the magnitudes of F1 and the resultant force. Set f = 30.
y F1 u
f
30 x F2
12 5 13
200 N
F3
260 N
52
254. Three forces act on the bracket. Determine the magnitude and direction u of F2 so that the resultant force is directed along the positive u axis and has a magnitude of 50 lb.
y F3
13 5 12
52 lb
F1 25 u F2
80 lb x
u
53
255. If F2 = 150 lb and u = 55, determine the magnitude and direction measured clockwise from the positive x axis of the resultant force of the three forces acting on the bracket.
y F3
13 5 12
52 lb
F1 25 u F2
80 lb x
u
*256. The three concurrent forces acting on the post produce a resultant force FR = 0. If F2 = 1 F1, and F1 is to 2 be 90 from F2 as shown, determine the required magnitude of F3 expressed in terms of F1 and the angle u.
y
F2
u
F3
x
F1
54
257. Determine the magnitude of force F so that the resultant force of the three forces is as small as possible. What is the magnitude of this smallest resultant force?
14 kN 45
F
30
8 kN
55
258. Express each of the three forces acting on the bracket in Cartesian vector form with respect to the x and y axes. Determine the magnitude and direction u of F1 so that the resultant force is directed along the positive x axis and has a magnitude of FR = 600 N.
y
F1 u 30 F2 F3 30 100 N
x
x 350 N
56
259. Determine the coordinate angle g for F2 and then express each force acting on the bracket as a Cartesian vector.
z
F1
450 N
45 30 45 x F2 600 N 60 y
57
*260. Determine the magnitude and coordinate direction angles of the resultant force acting on the bracket.
z
F1
450 N
45 30 45 x F2 600 N 60 y
58
261. Express each force acting on the pipe assembly in Cartesian vector form.
z
F1
600 lb
5 3 4
120 y
60 x
F2
400 lb
59
262. Determine the magnitude and direction of the resultant force acting on the pipe assembly.
z
F1
600 lb
5 3 4
120 y
60 x
F2
400 lb
60
263. The force F acts on the bracket within the octant shown. If F = 400 N, b = 60, and g = 45, determine the x, y, z components of F.
z g F
b a x y
61
*264. The force F acts on the bracket within the octant shown. If the magnitudes of the x and z components of F are Fx = 300 N and Fz = 600 N, respectively, and b = 60, determine the magnitude of F and its y component. Also, find the coordinate direction angles a and g.
z g F
b a x y
62
265. The two forces F1 and F2 acting at A have a resultant force of FR = 5 -100k6 lb. Determine the magnitude and coordinate direction angles of F2.
z
B 30 y A x F2 50
F1
60 lb
63
266. Determine the coordinate direction angles of the force F1 and indicate them on the figure.
B
z
30 y A x F2 50
F1
60 lb
267. The spur gear is subjected to the two forces caused by contact with other gears. Express each force as a Cartesian vector.
z 60
F2
180 lb
60
135 y
x
25 24 7
F1
50 lb
*268. The spur gear is subjected to the two forces caused by contact with other gears. Determine the resultant of the two forces and express the result as a Cartesian vector.
z 60
F2
180 lb
60
135 y
x
25 24 7
F1
50 lb
64
269. If the resultant force acting on the bracket is FR = 5 - 300i + 650j + 250k6 N, determine the magnitude and coordinate direction angles of F.
z
g a b
F
y x 30 45
F1
750 N
65
270. If the resultant force acting on the bracket is to be FR = 5800j6 N, determine the magnitude and coordinate direction angles of F.
z
g a b
F
y x 30 45
F1
750 N
66
271. If a = 120, b 6 90, g = 60, and F = 400 lb, determine the magnitude and coordinate direction angles of the resultant force acting on the hook.
z
F g a b 30 y F1 600 lb
4 5 3
x
67
*272. If the resultant force acting on the hook is FR = 5 -200i + 800j + 150k6 lb, determine the magnitude and coordinate direction angles of F.
z
F g a b 30 y F1 600 lb
4 5 3
x
68
273. The shaft S exerts three force components on the die D. Find the magnitude and coordinate direction angles of the resultant force. Force F2 acts within the octant shown.
z F3 200 N g2
5 4
60
F2
300 N
3
S y D
a2
60
F1 x
400 N
274. The mast is subjected to the three forces shown. Determine the coordinate direction angles a1, b 1, g1 of F1 so that the resultant force acting on the mast is FR = 5350i6 N.
a1 F3 300 N
z F1 g1
b1 y
x
F2
200 N
69
275. The mast is subjected to the three forces shown. Determine the coordinate direction angles a1, b 1, g1 of F1 so that the resultant force acting on the mast is zero.
a1 F3 300 N
z F1 g1
b1 y
x
F2
200 N
*276. Determine the magnitude and coordinate direction angles of F2 so that the resultant of the two forces acts along the positive x axis and has a magnitude of 500 N.
F2 g2
z
b2 a2 60 15 x F1 180 N y
70
277. Determine the magnitude and coordinate direction angles of F2 so that the resultant of the two forces is zero.
F2 g2
z
b2 a2 60 15 x F1 180 N y
71
278. If the resultant force acting on the bracket is directed along the positive y axis, determine the magnitude of the resultant force and the coordinate direction angles of F so that b 6 90.
z
g
F
500 N
b a 30 y x 30
F1
600 N
72
279. Specify the magnitude of F3 and its coordinate direction angles a3, b 3, g3 so that the resultant force FR = 59j6 kN.
z F2 g3 a3 30 F1 x 12 kN
13 5 12
10 kN
F3 b3
y
73
*280. If F3 = 9 kN, u = 30, and f = 45, determine the magnitude and coordinate direction angles of the resultant force acting on the ball-and-socket joint.
F1 60 30 10 kN
z
F2
8 kN
5 3
4
F3 f u
x
y
74
281. The pole is subjected to the force F, which has components acting along the x, y, z axes as shown. If the magnitude of F is 3 kN, b = 30, and g = 75, determine the magnitudes of its three components.
z Fz
g b a Fx x
F Fy y
282. The pole is subjected to the force F which has components Fx = 1.5 kN and Fz = 1.25 kN. If b = 75, determine the magnitudes of F and Fy.
z Fz
g b a Fx x
F Fy y
75
283. Three forces act on the ring. If the resultant force FR has a magnitude and direction as shown, determine the magnitude and the coordinate direction angles of force F3.
F2
z F3 110 N FR 120 N
F1
80 N
3
5 4
45 30
y
x
*284. Determine the coordinate direction angles of F1 and FR.
F2
z F3 110 N FR 120 N
F1
80 N
3
5 4
45 30
y
x
76
285. Two forces F1 and F2 act on the bolt. If the resultant force FR has a magnitude of 50 lb and coordinate direction angles a = 110 and b = 80, as shown, determine the magnitude of F2 and its coordinate direction angles.
z
g y 80 110 x F1 20 lb FR 50 lb F2
286. Determine the position vector r directed from point A to point B and the length of cord AB. Take z = 4 m.
3m
z 6m B z A y 2m x
77
287. If the cord AB is 7.5 m long, determine the coordinate position +z of point B
3m
z 6m B z A y 2m x
*288. Determine the distance between the end points A and B on the wire by first formulating a position vector from A to B and then determining its magnitude.
z A 1 in. 30 y 60 8 in.
3 in.
x B
2 in.
78
289. Determine the magnitude and coordinate direction angles of the resultant force acting at A.
4 ft
z
A
3 ft B 2.5 ft 3 ft
FB
600 lb FC 750 lb
4 ft x C 2 ft
79
290. Determine the magnitude and coordinate direction angles of the resultant force.
z 2m
A 500 N
600 N B
4m y 4m
8m x
C
80
291. Determine the magnitude and coordinate direction angles of the resultant force acting at A.
z
FB FC C
900 N A 600 N
6m 3m B 6m 45 4.5 m
y
x
81
*292. Determine the magnitude and coordinate direction angles of the resultant force.
z
C F2 81 lb F1 B 7 ft x 4 ft 3 ft 40 4 ft y A 100 lb
293. The chandelier is supported by three chains which are concurrent at point O. If the force in each chain has a magnitude of 60 lb, express each force as a Cartesian vector and determine the magnitude and coordinate direction angles of the resultant force.
z
O FB FC FA B 120 A 120 120 4 ft C y 6 ft
x
82
294. The chandelier is supported by three chains which are concurrent at point O. If the resultant force at O has a magnitude of 130 lb and is directed along the negative z axis, determine the force in each chain.
FB
z
O FC FA B 120 A 120 120 4 ft C y 6 ft
x
295. Express force F as a Cartesian vector; then determine its coordinate direction angles.
z
F
135 lb
A
10 ft 70 30
y 5 ft B 7 ft
x
83
*296. The tower is held in place by three cables. If the force of each cable acting on the tower is shown, determine the magnitude and coordinate direction angles a, b, g of the resultant force. Take x = 20 m, y = 15 m.
z D 600 N 400 N 24 m 800 N
16 m C 18 m y A x O B x
4m 6m y
84
297. The door is held opened by means of two chains. If the tension in AB and CD is FA = 300 N and FC = 250 N, respectively, express each of these forces in Cartesian vector form.
z
C
2.5 m FC A FA 300 N 250 N
1.5 m
30 1m B
D 0.5 m
y x
298. The guy wires are used to support the telephone pole. Represent the force in each wire in Cartesian vector form. Neglect the diameter of the pole.
z
B FB 175 N 4m D 3m x
1.5 m A FA 250 N 4m C 1m y
2m
85
299. Two cables are used to secure the overhang boom in position and support the 1500-N load. If the resultant force is directed along the boom from point A towards O, determine the magnitudes of the resultant force and forces FB and FC. Set x = 3 m and z = 2 m.
z x C 2m B 3m
z 6 m FC
FB A y
x
1500 N
86
*2100. Two cables are used to secure the overhang boom in position and support the 1500-N load. If the resultant force is directed along the boom from point A towards O, determine the values of x and z for the coordinates of point C and the magnitude of the resultant force. Set FB = 1610 N and FC = 2400 N.
z x C 2m B 3m
z 6 m FC
FB A y
x
1500 N
87
2101. The cable AO exerts a force on the top of the pole of F = 5 -120i - 90j - 80k6 lb. If the cable has a length of 34 ft, determine the height z of the pole and the location (x, y) of its base.
z A F
z O x y x y
88
2102. If the force in each chain has a magnitude of 450 lb, determine the magnitude and coordinate direction angles of the resultant force.
z
D
FC
FB
FA
7 ft
B 120 120 C x 120 3 ft A
y
89
2103. If the resultant of the three forces is FR = 5 -900k6 lb, determine the magnitude of the force in each chain.
z
D
FC
FB
FA
7 ft
B 120 120 C x 120 3 ft A
y
90
*2104. The antenna tower is supported by three cables. If the forces of these cables acting on the antenna are FB = 520 N, FC = 680 N, and FD = 560 N, determine the magnitude and coordinate direction angles of the resultant force acting at A.
24 m
z A FB FD FC 8m B 10 m O 16 m x 18 m C D
12 m y
91
2105. If the force in each cable tied to the bin is 70 lb, determine the magnitude and coordinate direction angles of the resultant force.
z
E FA FB A x 2 ft 2 ft B 3 ft FC FD D C 3 ft y 6 ft
92
2106. If the resultant of the four forces is FR = 5 - 360k6 lb, determine the tension developed in each cable. Due to symmetry, the tension in the four cables is the same.
FA
z
E FC FB A x 2 ft 2 ft B 3 ft FD D C 3 ft y 6 ft
93
2107. The pipe is supported at its end by a cord AB. If the cord exerts a force of F = 12 lb on the pipe at A, express this force as a Cartesian vector.
z B
6 ft
F x
3 ft
12 lb
5 ft y
A
20
*2108. The load at A creates a force of 200 N in wire AB. Express this force as a Cartesian vector, acting on A and directed towards B.
120
z
30 1m 120
y B
2m x F A 200 N
94
2109. The cylindrical plate is subjected to the three cable forces which are concurrent at point D. Express each force which the cables exert on the plate as a Cartesian vector, and determine the magnitude and coordinate direction angles of the resultant force.
FC 5 kN
z D
3m FB 8 kN
C 45
B
30 y 0.75 m
A x
FA
6 kN
95
2110. The cable attached to the shear-leg derrick exerts a force on the derrick of F = 350 lb. Express this force as a Cartesian vector.
35 ft
z
A
F 30 50 ft x
350 lb
y B
2111. Given the three vectors A, B, and D, show that A # (B + D) = (A # B) + (A # D).
96
*2112. Determine the projected component of the force FAB = 560 N acting along cable AC. Express the result as a Cartesian vector.
C
z 1.5 m 1.5 m B 1m
3m
FAB A
560 N
x
y 3m
97
2113. Determine the magnitudes of the components of force F = 56 N acting along and perpendicular to line AO.
z D 1m C 1m O x B 3m 1.5 m y A F 56 N
98
2114. Determine the length of side BC of the triangular plate. Solve the problem by finding the magnitude of rBC; then check the result by first finding q , rAB, and rAC and then using the cosine law.
z
3m
B
4m A 1m 1m C u y 3m
5m x
99
2115. Determine the magnitudes of the components of F = 600 N acting along and perpendicular to segment DE of the pipe assembly.
2m
z A B 2m x 2m D 3m E 2m C F 600 N y
100
*2116. Two forces act on the hook. Determine the angle u between them. Also, what are the projections of F1 and F2 along the y axis?
z
F1 45
600 N
60 120 u y x F2 {120i + 90j 80k}N
2117. Two forces act on the hook. Determine the magnitude of the projection of F2 along F1.
z
F1 45
600 N
60 120 u y x F2 {120i + 90j 80k}N
101
2118. Determine the projection of force F = 80 N along line BC. Express the result as a Cartesian vector.
z A F 80 N D
E
1.5 m x
C F 2m
B 2m
1.5 m y
2m 2m
102
2119. The clamp is used on a jig. If the vertical force acting on the bolt is F = { -500k} N, determine the magnitudes of its components F1 and F2 which act along the OA axis and perpendicular to it.
z
A
O x 40 mm
40 mm y 20 mm
F
{ 500 k} N
103
*2120. Determine the magnitude of the projected component of force FAB acting along the z axis.
FAC FAB 600 lb
z A 36 ft
700 lb D
18 ft O B 12 ft x 12 ft C 12 ft
30
y
104
2121. Determine the magnitude of the projected component of force FAC acting along the z axis.
FAC FAB 600 lb
z A 36 ft
700 lb D
18 ft O B 12 ft x 12 ft C 12 ft
30
y
105
2122. Determine the projection of force F = 400 N acting along line AC of the pipe assembly. Express the result as a Cartesian vector.
B
z F 400 N 45 30 A 3m x 4m y
C
106
2123. Determine the magnitudes of the components of force F = 400 N acting parallel and perpendicular to segment BC of the pipe assembly.
B
z F 400 N 45 30 A 3m x 4m y
C
107
*2124. Cable OA is used to support column OB. Determine the angle u it makes with beam OC.
O
z D 30 f C x 8m u 8m 4m y
B
A
2125. Cable OA is used to support column OB. Determine the angle f it makes with beam OD.
O
z D 30 f C x 8m u 8m 4m y
B
A
108
2126. The cables each exert a force of 400 N on the post. Determine the magnitude of the projected component of F1 along the line of action of F2.
z F1 400 N
35
120 u 45 60
y
20
x
F2
400 N
2127. Determine the angle u between the two cables attached to the post.
z F1 400 N
35
120 u 45 60
y
20
x
F2
400 N
109
*2128. A force of F = 80 N is applied to the handle of the wrench. Determine the angle u between the tail of the force and the handle AB.
z
F u
80 N 30 45 A
B
300 mm y
x
500 mm
2129. Determine the angle u between cables AB and AC.
8 ft C 12 ft
z 3 ft B 8 ft y u A F 15 ft
x
110
2130. If F has a magnitude of 55 lb, determine the magnitude of its projected components acting along the x axis and along cable AC.
C 12 ft
z 8 ft 3 ft B 8 ft y u A F 15 ft
x
2131. Determine the magnitudes of the projected components of the force F = 300 N acting along the x and y axes.
z
30 A 30
F
300 N
300 mm O 300 mm x 300 mm y
111
*2132. Determine the magnitude of the projected component of the force F = 300 N acting along line OA.
30 z A 30
F
300 N
300 mm O 300 mm x 300 mm y
112
2133. Two cables exert forces on the pipe. Determine the magnitude of the projected component of F1 along the line of action of F2.
z
F2 60
25 lb
u x
60
30
30
y
F1
30 lb
2134. Determine the angle u between the two cables attached to the pipe.
z
F2 60
25 lb
u x
60
30
30
y
F1
30 lb
113
2135. force.
Determine the x and y components of the 700-lb
700 lb
y
60
30
x
114
*2136. Determine the magnitude of the projected component of the 100-lb force acting along the axis BC of the pipe.
B
z A u x 4 ft 2 ft C 6 ft F 100 lb D y 3 ft 8 ft
2137. Determine the angle u between pipe segments BA and BC.
B
z A u x 4 ft 2 ft C 6 ft F 100 lb D y 3 ft 8 ft
115
2138. Determine the magnitude and direction of the resultant FR = F1 + F2 + F3 of the three forces by first finding the resultant F = F1 + F3 and then forming FR = F + F2. Specify its direction measured counterclockwise from the positive x axis.
y F1 80 N F2 75 N F3 30 50 N
30 45 x
116
2139. Determine the design angle u (u < 90) between the two struts so that the 500-lb horizontal force has a component of 600 lb directed from A toward C. What is the component of force acting along member BA?
B
C
20 u 500 lb A
117
*2140. Determine the magnitude and direction of the smallest force F3 so that the resultant force of all three forces has a magnitude of 20 lb.
F2
5 3
10 lb
4
F3 u F1 5 lb
118
2141. Resolve the 250-N force into components acting along the u and v axes and determine the magnitudes of these components.
u 20 250 N 40 v
2142. Cable AB exerts a force of 80 N on the end of the 3-m-long boom OA. Determine the magnitude of the projection of this force along the boom.
z B
4m
O 60 80 N 3m A x
y
119
2143. The three supporting cables exert the forces shown on the sign. Represent each force as a Cartesian vector.
C 2m
z
E
2m B
FE FC 400 N FB
350 N
3m
400 N
D A x 3m
2m
y
120
31. Determine the force in each cord for equilibrium of the 200-kg crate. Cord BC remains horizontal due to the roller at C, and AB has a length of 1.5 m. Set y = 0.75 m.
2m A y
C
B
121
32. If the 1.5-m-long cord AB can withstand a maximum force of 3500 N, determine the force in cord BC and the distance y so that the 200-kg crate can be supported.
2m A y
C
B
122
33. If the mass of the girder is 3 Mg and its center of mass is located at point G, determine the tension developed in cables AB, BC, and BD for equilibrium.
FAB A B 45 C G 30 D
123
*34. If cables BD and BC can withstand a maximum tensile force of 20 kN, determine the maximum mass of the girder that can be suspended from cable AB so that neither cable will fail. The center of mass of the girder is located at point G.
45 C
FAB A B 30 D G
124
35. The members of a truss are connected to the gusset plate. If the forces are concurrent at point O, determine the magnitudes of F and T for equilibrium. Take u = 30.
A 8 kN u 45 D B T C 5 kN O
F
36. The gusset plate is subjected to the forces of four members. Determine the force in member B and its proper orientation u for equilibrium. The forces are concurrent at point O. Take F = 12 kN.
A 8 kN u
O
45 D B T C 5 kN
F
125
37. The towing pendant AB is subjected to the force of 50 kN exerted by a tugboat. Determine the force in each of the bridles, BC and BD, if the ship is moving forward with constant velocity.
D
C 30
20
B
A 50 kN
126
*38. Members AC and AB support the 300-lb crate. Determine the tensile force developed in each member.
C
3 ft
4 ft B
4 ft
A
127
39. If members AC and AB can support a maximum tension of 300 lb and 250 lb, respectively, determine the largest weight of the crate that can be safely supported.
C
3 ft
4 ft B
4 ft
A
128
310. The members of a truss are connected to the gusset plate. If the forces are concurrent at point O, determine the magnitudes of F and T for equilibrium. Take u = 90.
y
9 kN F A
5 3 B 4
O
u
x
C T
311. The gusset plate is subjected to the forces of three members. Determine the tension force in member C and its angle u for equilibrium. The forces are concurrent at point O. Take F = 8 kN.
y
9 kN F A
5 3 B 4
O
u
x
C T
129
*312. If block B weighs 200 lb and block C weighs 100 lb, determine the required weight of block D and the angle u for equilibrium.
u A B D C 30
130
313. If block D weighs 300 lb and block B weighs 275 lb, determine the required weight of block C and the angle u for equilibrium.
u A B D C 30
131
314. Determine the stretch in springs AC and AB for equilibrium of the 2-kg block. The springs are shown in the equilibrium position.
3m C
4m B
3m
kAC
20 N/m kAB 30 N/m
A
D
315. The unstretched length of spring AB is 3 m. If the block is held in the equilibrium position shown, determine the mass of the block at D.
3m C
4m B
3m
kAC
20 N/m kAB 30 N/m
A
D
132
*316. Determine the tension developed in wires CA and CB required for equilibrium of the 10-kg cylinder. Take u = 40.
A u C
B
30
133
317. If cable CB is subjected to a tension that is twice that of cable CA, determine the angle u for equilibrium of the 10-kg cylinder. Also, what are the tensions in wires CA and CB?
A u C
B
30
134
318. Determine the forces in cables AC and AB needed to hold the 20-kg ball D in equilibrium. Take F = 300 N and d = 1 m.
1.5 m
B
C d A 2m D F
135
319. The ball D has a mass of 20 kg. If a force of F = 100 N is applied horizontally to the ring at A, determine the dimension d so that the force in cable AC is zero.
1.5 m
B
C d A 2m D F
136
*320. Determine the tension developed in each wire used to support the 50-kg chandelier.
A 30 B 45 D 30 C
137
321. If the tension developed in each of the four wires is not allowed to exceed 600 N, determine the maximum mass of the chandelier that can be supported.
A 30 B 45 D 30 C
138
322. A vertical force P = 10 lb is applied to the ends of the 2-ft cord AB and spring AC. If the spring has an unstretched length of 2 ft, determine the angle u for equilibrium. Take k = 15 lb>ft.
2 ft u
2 ft
B
C
k A
P
139
323. Determine the unstretched length of spring AC if a force P = 80 lb causes the angle u = 60 for equilibrium. Cord AB is 2 ft long. Take k = 50 lb>ft.
B
2 ft u
2 ft
C
k A
P
140
*324. If the bucket weighs 50 lb, determine the tension developed in each of the wires.
C B A
4 3 5
30 D 30 E
141
325. Determine the maximum weight of the bucket that the wire system can support so that no single wire develops a tension exceeding 100 lb.
B A
4 3 5
C
30 D 30 E
142
326. Determine the tensions developed in wires CD, CB, and BA and the angle u required for equilibrium of the 30-lb cylinder E and the 60-lb cylinder F.
D 30 C u B 45
A
E F
143
327. If cylinder E weighs 30 lb and u = 15, determine the weight of cylinder F.
D 30 C u B 45
A
E F
144
*328. Two spheres A and B have an equal mass and are electrostatically charged such that the repulsive force acting between them has a magnitude of 20 mN and is directed along line AB. Determine the angle u, the tension in cords AC and BC, and the mass m of each sphere.
u
C
30 B
20 mN 20 mN A 30
329. The cords BCA and CD can each support a maximum load of 100 lb. Determine the maximum weight of the crate that can be hoisted at constant velocity and the angle u for equilibrium. Neglect the size of the smooth pulley at C.
D
C
u
13 5
12
A
B
145
330. The springs on the rope assembly are originally unstretched when u = 0. Determine the tension in each rope when F = 90 lb. Neglect the size of the pulleys at B and D.
2 ft B
2 ft
A k 30 lb/ft
D
k
30 lb/ft
F C E
146
331. The springs on the rope assembly are originally stretched 1 ft when u = 0. Determine the vertical force F that must be applied so that u = 30.
2 ft B
2 ft
A k 30 lb/ft
D
k
30 lb/ft
F C E
147
*332. Determine the magnitude and direction u of the equilibrium force FAB exerted along link AB by the tractive apparatus shown. The suspended mass is 10 kg. Neglect the size of the pulley at A.
75 A 45 B FAB u
148
333. The wire forms a loop and passes over the small pulleys at A, B, C, and D. If its end is subjected to a force of P = 50 N, determine the force in the wire and the magnitude of the resultant force that the wire exerts on each of the pulleys.
B
A 30 30 D
C
P
149
334. The wire forms a loop and passes over the small pulleys at A, B, C, and D. If the maximum resultant force that the wire can exert on each pulley is 120 N, determine the greatest force P that can be applied to the wire as shown.
A 30 30
B
C
D
P
150
335. The picture has a weight of 10 lb and is to be hung over the smooth pin B. If a string is attached to the frame at points A and C, and the maximum force the string can support is 15 lb, determine the shortest string that can be safely used.
B A C
9 in.
9 in.
151
*336. The 200-lb uniform tank is suspended by means of a 6-ft-long cable, which is attached to the sides of the tank and passes over the small pulley located at O. If the cable can be attached at either points A and B or C and D, determine which attachment produces the least amount of tension in the cable. What is this tension?
F
O B C D A 2 ft 2 ft 2 ft 1 ft
152
337. The 10-lb weight is supported by the cord AC and roller and by the spring that has a stiffness of k = 10 lb>in. and an unstretched length of 12 in. Determine the distance d to where the weight is located when it is in equilibrium.
d
12 in. u B
k
C
A
153
338. The 10-lb weight is supported by the cord AC and roller and by a spring. If the spring has an unstretched length of 8 in. and the weight is in equilibrium when d = 4 in., determine the stiffness k of the spring.
d
12 in. u B
k
C
A
154
339. A "scale" is constructed with a 4-ft-long cord and the 10-lb block D. The cord is fixed to a pin at A and passes over two small pulleys at B and C. Determine the weight of the suspended block at B if the system is in equilibrium.
1 ft
A
C
1.5 ft
D B
*340. The spring has a stiffness of k = 800 N>m and an unstretched length of 200 mm. Determine the force in cables BC and BD when the spring is held in the position shown.
C
400 mm A k 800 N/m B
300 mm
D 500 mm 400 mm
155
341. A continuous cable of total length 4 m is wrapped around the small pulleys at A, B, C, and D. If each spring is stretched 300 mm, determine the mass m of each block. Neglect the weight of the pulleys and cords. The springs are unstretched when d = 2 m.
B d A C
k
500 N/m
D
k
500 N/m
342. Determine the mass of each of the two cylinders if they cause a sag of s = 0.5 m when suspended from the rings at A and B. Note that s = 0 when the cylinders are removed.
2m
1m
2m
C 1.5 m
D
s
k
100 N/m A B
k
100 N/m
156
343. The pail and its contents have a mass of 60 kg. If the cable BAC is 15 m long, determine the distance y to the pulley at A for equilibrium. Neglect the size of the pulley.
C 2m B y A
10 m
157
*344. A scale is constructed using the 10-kg mass, the 2-kg pan P, and the pulley and cord arrangement. Cord BCA is 2 m long. If s = 0.75 m, determine the mass D in the pan. Neglect the size of the pulley.
1.5 m A
1.5
C
0 s
m
B D P
158
345. Determine the tension in the cables in order to support the 100-kg crate in the equilibrium position shown.
2m C
z
D A 1m 2m 2m y
2.5 m B x
159
346. Determine the maximum mass of the crate so that the tension developed in any cable does not exceeded 3 kN.
2m C
z
D A 1m 2m 2m y
2.5 m B x
160
347. The shear leg derrick is used to haul the 200-kg net of fish onto the dock. Determine the compressive force along each of the legs AB and CB and the tension in the winch cable DB. Assume the force in each leg acts along its axis.
z 5.6 m 4m B
D C A x 2m 2m y 4m
161
*348. Determine the tension developed in cables AB, AC, and AD required for equilibrium of the 300-lb crate.
z C 1 ft 2 ft A 3 ft D x 2 ft
B 2 ft
1 ft
2 ft y
162
349. Determine the maximum weight of the crate so that the tension developed in any cable does not exceed 450 lb.
z C 1 ft 2 ft A 3 ft D x 2 ft
B 2 ft
1 ft
2 ft y
163
350. Determine the force in each cable needed to support the 3500-lb platform. Set d = 2 ft.
z 3500 lb A
10 ft
D B 3 ft x 3 ft d
2 ft
4 ft C 4 ft y
164
351. Determine the force in each cable needed to support the 3500-lb platform. Set d = 4 ft.
z 3500 lb A
10 ft
D B 3 ft x 3 ft d
2 ft
4 ft C 4 ft y
165
*352. Determine the force in each of the three cables needed to lift the tractor which has a mass of 8 Mg.
z
A 3m
D B 1.25 m 1.25 m x C 2m 1m y
166
353. Determine the force acting along the axis of each of the three struts needed to support the 500-kg block.
D C
z
A
B 2.5 m 2m x 1.25 m 3m 0.75 m y
167
354. If the mass of the flowerpot is 50 kg, determine the tension developed in each wire for equilibrium. Set x = 1.5 m and z = 2 m.
z 2m x 3m D
C
z
x
6m
A
B y
168
355. If the mass of the flowerpot is 50 kg, determine the tension developed in each wire for equilibrium. Set x = 2 m and z = 1.5 m.
z 2m x 3m D
C
z
x
6m
A
B y
169
*356. The ends of the three cables are attached to a ring at A and to the edge of a uniform 150-kg plate. Determine the tension in each of the cables for equilibrium.
z
A
2m C
10 m D
12 m 2m
4m y
B 2m 6m x 6m
6m
357. The ends of the three cables are attached to a ring at A and to the edge of the uniform plate. Determine the largest mass the plate can have if each cable can support a maximum tension of 15 kN.
z
A
2m C
10 m D
12 m 2m
4m y
B 2m 6m x 6m
6m
170
358. Determine the tension developed in cables AB, AC, and AD required for equilibrium of the 75-kg cylinder.
C 2m 1m 3m D 4m x 1.5 m
z
B
2m
3m
1m y A
171
359. If each cable can withstand a maximum tension of 1000 N, determine the largest mass of the cylinder for equilibrium.
C 2m 1m 3m D 4m x 1.5 m
z
B
2m
3m
1m y A
172
*360. The 50-kg pot is supported from A by the three cables. Determine the force acting in each cable for equilibrium. Take d = 2.5 m.
z 2m C 2m D 3m
6m A B 6m d x y
173
361. Determine the height d of cable AB so that the force in cables AD and AC is one-half as great as the force in cable AB. What is the force in each cable for this case? The flower pot has a mass of 50 kg.
z 2m C 2m D 3m
6m A B 6m d x y
174
362. A force of F = 100 lb holds the 400-lb crate in equilibrium. Determine the coordinates (0, y, z) of point A if the tension in cords AC and AB is 700 lb each.
C 4 ft
z 5 ft 5 ft B
A x y z y F
175
363. If the maximum allowable tension in cables AB and AC is 500 lb, determine the maximum height z to which the 200-lb crate can be lifted. What horizontal force F must be applied? Take y = 8 ft.
C 4 ft
z 5 ft 5 ft B
A x y z y F
176
*364. The thin ring can be adjusted vertically between three equally long cables from which the 100-kg chandelier is suspended. If the ring remains in the horizontal plane and z = 600 mm, determine the tension in each cable.
D x z
z 0.5 m 120 120 B y
C 120
A
177
365. The thin ring can be adjusted vertically between three equally long cables from which the 100-kg chandelier is suspended. If the ring remains in the horizontal plane and the tension in each cable is not allowed to exceed 1 kN, determine the smallest allowable distance z required for equilibrium.
x
z 0.5 m 120 120 B y
C 120 D
z A
178
366. The bucket has a weight of 80 lb and is being hoisted using three springs, each having an unstretched length of l0 = 1.5 ft and stiffness of k = 50 lb>ft. Determine the vertical distance d from the rim to point A for equilibrium.
80 lb
A d 120 120 B 1.5 ft 120 C
D
179
367. Three cables are used to support a 900-lb ring. Determine the tension in each cable for equilibrium.
z
F
A
4 ft D 120 120 3 ft 120 B
y
C x
180
*368. The three outer blocks each have a mass of 2 kg, and the central block E has a mass of 3 kg. Determine the sag s for equilibrium of the system.
A
1m 30 60 30 1m
B
C s
D
E
181
369. Determine the angle u such that an equal force is developed in legs OB and OC. What is the force in each leg if the force is directed along the axis of each leg? The force F lies in the x-y plane. The supports at A, B, C can exert forces in either direction along the attached legs.
x
z
O u y 10 ft 120 120 C 5 ft A
F = 100 lb
B 120
182
370. The 500-lb crate is hoisted using the ropes AB and AC. Each rope can withstand a maximum tension of 2500 lb before it breaks. If AB always remains horizontal, determine the smallest angle u to which the crate can be hoisted.
C
u F
A
B
371. The members of a truss are pin connected at joint O. Determine the magnitude of F1 and its angle u for equilibrium. Set F2 = 6 kN.
y 5 kN F2 70 30 x O u
5 4
3
7 kN
F1
183
*372. The members of a truss are pin connected at joint O. Determine the magnitudes of F1 and F2 for equilibrium. Set u = 60.
y 5 kN F2 70 30 x O u
5 4
3
7 kN
F1
184
373. Two electrically charged pith balls, each having a mass of 0.15 g, are suspended from light threads of equal length. Determine the magnitude of the horizontal repulsive force, F, acting on each ball if the measured distance between them is r = 200 mm.
150 mm
50 mm 150 mm
A
F
F
B
r
200 mm
185
374. The lamp has a mass of 15 kg and is supported by a pole AO and cables AB and AC. If the force in the pole acts along its axis, determine the forces in AO, AB, and AC for equilibrium.
z A B
4m 6m O 2m 1.5 m 1.5 m C y
x
186
375. Determine the magnitude of P and the coordinate direction angles of F3 required for equilibrium of the particle. Note that F3 acts in the octant shown.
z ( 1 ft, 7 ft, 4 ft) F1 360 lb P F2 120 lb 20 y
F4 F3 200 lb x
300 lb
187
*376. The ring of negligible size is subjected to a vertical force of 200 lb. Determine the longest length l of cord AC such that the tension acting in AC is 160 lb. Also, what is the force acting in cord AB? Hint: Use the equilibrium condition to determine the required angle u for attachment, then determine l using trigonometry applied to ABC.
C
u l A
40 2 ft
B
200 lb
377. Determine the magnitudes of F1, F2, and F3 for equilibrium of the particle.
z F1
60 F3 60
135
5 4 3
800 lb y
P 200 lb
F2 x
188
378. Determine the force in each cable needed to support the 500-lb load.
z
D
8 ft y 6 ft B C 2 ft 2 ft A x 6 ft
379. The joint of a space frame is subjected to four member forces. Member OA lies in the x y plane and member OB lies in the y z plane. Determine the forces acting in each of the members required for equilibrium of the joint.
F3
z
A O 45
F1 y
B 40 F2 x 200 lb
189
41. If A, B, and D are given vectors, prove the distributive law for the vector cross product, i.e., A : (B + D) = (A : B) + (A : D).
190
42. Prove the triple A # B : C = A : B # C.
scalar
product
identity
191
43. Given the three nonzero vectors A, B, and C, show that if A # (B : C) = 0, the three vectors must lie in the same plane.
*44. Two men exert forces of F = 80 lb and P = 50 lb on the ropes. Determine the moment of each force about A. Which way will the pole rotate, clockwise or counterclockwise?
P
6 ft
F 45
3 5 4
B
12 ft
C A
45. If the man at B exerts a force of P = 30 lb on his rope, determine the magnitude of the force F the man at C must exert to prevent the pole from rotating, i.e., so the resultant moment about A of both forces is zero.
P
6 ft
F 45
3 5 4
B
12 ft
C A
192
46. If u = 45, determine the moment produced by the 4-kN force about point A.
3m A 0.45 m u 4 kN
193
47. If the moment produced by the 4-kN force about point A is 10 kN # m clockwise, determine the angle u, where 0 ... u ... 90.
3m A 0.45 m u 4 kN
194
*48. The handle of the hammer is subjected to the force of F = 20 lb. Determine the moment of this force about the point A.
F 30
5 in. 18 in. A B
195
49. In order to pull out the nail at B, the force F exerted on the handle of the hammer must produce a clockwise moment of 500 lb # in. about point A. Determine the required magnitude of force F.
F 30
5 in. 18 in. A B
196
410. The hub of the wheel can be attached to the axle either with negative offset (left) or with positive offset (right). If the tire is subjected to both a normal and radial load as shown, determine the resultant moment of these loads about point O on the axle for both cases.
0.05 m O O
0.05 m
0.4 m
0.4 m
800 N 4 kN Case 1 4 kN Case 2
800 N
197
411. The member is subjected to a force of F = 6 kN. If u = 45, determine the moment produced by F about point A.
1.5 m u F 6m 6 kN
A
198
*412. Determine the angle u (0 ... u ... 180) of the force F so that it produces a maximum moment and a minimum moment about point A. Also, what are the magnitudes of these maximum and minimum moments?
1.5 m u F 6m 6 kN
A
199
413. Determine the moment produced by the force F about point A in terms of the angle u. Plot the graph of MA versus u, where 0 ... u ... 180.
1.5 m u F 6m 6 kN
A
200
414. Serious neck injuries can occur when a football player is struck in the face guard of his helmet in the manner shown, giving rise to a guillotine mechanism. Determine the moment of the knee force P = 50 lb about point A. What would be the magnitude of the neck force F so that it gives the counterbalancing moment about A?
2 in. 60 A P 50 lb 4 in. F 6 in. 30
415. The Achilles tendon force of Ft = 650 N is mobilized when the man tries to stand on his toes. As this is done, each of his feet is subjected to a reactive force of Nf = 400 N. Determine the resultant moment of Ft and Nf about the ankle joint A.
Ft
5
A
200 mm
65 mm
100 mm
Nf
400 N
201
*416. The Achilles tendon force Ft is mobilized when the man tries to stand on his toes. As this is done, each of his feet is subjected to a reactive force of Nt = 400 N. If the resultant moment produced by forces Ft and Nt about the ankle joint A is required to be zero, determine the magnitude of Ft.
Ft
5
A
200 mm
65 mm
100 mm
Nf
400 N
417. The two boys push on the gate with forces of FA = 30 lb and as shown. Determine the moment of each force about C. Which way will the gate rotate, clockwise or counterclockwise? Neglect the thickness of the gate.
6 ft C B
3 ft
4
FA
5
A 60 FB
3
418. Two boys push on the gate as shown. If the boy at B exerts a force of FB = 30 lb, determine the magnitude of the force FA the boy at A must exert in order to prevent the gate from turning. Neglect the thickness of the gate.
6 ft C B
3 ft
4
FA
5
A 60 FB
3
202
419. The tongs are used to grip the ends of the drilling pipe P. Determine the torque (moment) MP that the applied force F = 150 lb exerts on the pipe about point P as a function of u. Plot this moment MP versus u for 0 ... u ... 90.
F u
P
6 in.
MP 43 in.
*420. The tongs are used to grip the ends of the drilling pipe P. If a torque (moment) of MP = 800 lb # ft is needed at P to turn the pipe, determine the cable force F that must be applied to the tongs. Set u = 30.
F u
P
6 in.
MP 43 in.
203
421. Determine the direction u for 0 ... u ... 180 of the force F so that it produces the maximum moment about point A. Calculate this moment.
F
u
400 N
2m
A 3m
204
422. Determine the moment of the force F about point A as a function of u. Plot the results of M (ordinate) versus u (abscissa) for 0 ... u ... 180.
F
u
400 N
2m
A 3m
423. Determine the minimum moment produced by the force F about point A. Specify the angle u (0 ... u ... 180).
F
u
400 N
2m
A 3m
205
*424. In order to raise the lamp post from the position shown, force F is applied to the cable. If F = 200 lb, determine the moment produced by F about point A.
B
F
20 ft 75
A
C 10 ft
206
425. In order to raise the lamp post from the position shown, the force F on the cable must create a counterclockwise moment of 1500 lb # ft about point A. Determine the magnitude of F that must be applied to the cable.
F
B
20 ft 75
A
C 10 ft
207
426. The foot segment is subjected to the pull of the two plantarflexor muscles. Determine the moment of each force about the point of contact A on the ground.
F2 F1 20 lb 30 70
30 lb
60
4 in.
A 1 in. 3.5 in.
427. The 70-N force acts on the end of the pipe at B. Determine (a) the moment of this force about point A, and (b) the magnitude and direction of a horizontal force, applied at C, which produces the same moment. Take u = 60.
A
0.9 m 70 N
u C 0.3 m 0.7 m B
208
*428. The 70-N force acts on the end of the pipe at B. Determine the angles u 10 ... u ... 1802 of the force that will produce maximum and minimum moments about point A. What are the magnitudes of these moments?
A
0.9 m 70 N
u C 0.3 m 0.7 m B
429. Determine the moment of each force about the bolt located at A. Take FB = 40 lb, FC = 50 lb.
209
430. If FB = 30 lb and FC = 45 lb, determine the resultant moment about the bolt located at A.
431. The rod on the power control mechanism for a business jet is subjected to a force of 80 N. Determine the moment of this force about the bearing at A.
80 N
20
150 mm
60 A
*432. The towline exerts a force of P = 4 kN at the end of the 20-m-long crane boom. If u = 30, determine the placement x of the hook at A so that this force creates a maximum moment about point O. What is this moment?
20 m O u 1.5 m x
B P 4 kN
A
210
433. The towline exerts a force of P = 4 kN at the end of the 20-m-long crane boom. If x = 25 m, determine the position u of the boom so that this force creates a maximum moment about point O. What is this moment?
20 m O u 1.5 m x
B P 4 kN
A
211
434. In order to hold the wheelbarrow in the position shown, force F must produce a counterclockwise moment of 200 N # m about the axle at A. Determine the required magnitude of force F.
B 0.65 m G 0.5 m A
30 F
1.2 m 0.3 m
212
435. The wheelbarrow and its contents have a mass of 50 kg and a center of mass at G. If the resultant moment produced by force F and the weight about point A is to be zero, determine the required magnitude of force F.
B 0.65 m G 0.5 m A
30 F
1.2 m 0.3 m
213
*436. The wheelbarrow and its contents have a center of mass at G. If F = 100 N and the resultant moment produced by force F and the weight about the axle at A is zero, determine the mass of the wheelbarrow and its contents.
B 0.65 m G 0.5 m A
30 F
1.2 m 0.3 m
214
437. Determine the moment produced by F1 about point O. Express the result as a Cartesian vector.
z 2 ft 3 ft x F2 { 10i 30j 50k} lb A O F1 { 20i 10j 30k} lb 1 ft y
2 ft
215
438. Determine the moment produced by F2 about point O. Express the result as a Cartesian vector.
z 2 ft 3 ft x F2 { 10i 30j 50k} lb A O F1 { 20i 10j 30k} lb 1 ft y
2 ft
216
439. Determine the resultant moment produced by the two forces about point O. Express the result as a Cartesian vector.
z 2 ft 3 ft x F2 { 10i 30j 50k} lb A O F1 { 20i 10j 30k} lb 1 ft y
2 ft
217
*440. Determine the moment produced by force FB about point O. Express the result as a Cartesian vector.
A FC 420 N
z
6m
FB
780 N
2m C 3m x O B y 2.5 m
218
441. Determine the moment produced by force FC about point O. Express the result as a Cartesian vector.
A FC 420 N
z
6m
FB
780 N
2m C 3m x O B y 2.5 m
219
442. Determine the resultant moment produced by forces FB and FC about point O. Express the result as a Cartesian vector.
6m FC 420 N A
z
FB
780 N
2m C 3m x O B y 2.5 m
220
443. Determine the moment produced by each force about point O located on the drill bit. Express the results as Cartesian vectors.
z 600 mm FA 300 mm O 150 mm x FB { 50i 120j 60k} N A B { 40i 100j 60k} N y
150 mm
221
*444. A force of F = 56i - 2j + 1k6 kN produces a moment of MO = 54i + 5j - 14k6 kN # m about the origin of coordinates, point O. If the force acts at a point having an x coordinate of x = 1 m, determine the y and z coordinates.
445. The pipe assembly is subjected to the 80-N force. Determine the moment of this force about point A.
z
A 400 mm B x 300 mm 200 mm C 250 mm 40 30 F 80 N y
222
446. The pipe assembly is subjected to the 80-N force. Determine the moment of this force about point B.
z
A 400 mm B x 300 mm 200 mm C 250 mm 40 30 F 80 N y
447. The force F = 56i + 8j + 10k6 N creates a moment about point O of MO = 5 -14i + 8j + 2k6 N # m. If the force passes through a point having an x coordinate of 1 m, determine the y and z coordinates of the point. Also, realizing that MO = Fd, determine the perpendicular distance d from point O to the line of action of F.
z
P
MO d z
F
O
1m y x
y
223
*448. Force F acts perpendicular to the inclined plane. Determine the moment produced by F about point A. Express the result as a Cartesian vector.
3m
z A 3m F 400 N
B x 4m
C y
224
449. Force F acts perpendicular to the inclined plane. Determine the moment produced by F about point B. Express the result as a Cartesian vector.
3m
z A 3m F 400 N
B x 4m
C y
225
450. A 20-N horizontal force is applied perpendicular to the handle of the socket wrench. Determine the magnitude and the coordinate direction angles of the moment created by this force about point O.
75 mm
z
200 mm
A 20 N
O 15 x y
226
451. Determine the moment produced by force F about the diagonal AF of the rectangular block. Express the result as a Cartesian vector.
z A C O 3m F D
F
{ 6i
3j
10k} N
B 1.5 m G 3m y
x
227
*452. Determine the moment produced by force F about the diagonal OD of the rectangular block. Express the result as a Cartesian vector.
z A C O 3m F D
F
{ 6i
3j
10k} N
B 1.5 m G 3m y
x
228
453. The tool is used to shut off gas valves that are difficult to access. If the force F is applied to the handle, determine the component of the moment created about the z axis of the valve.
z
0.25 m
F
{ 60i
20j
15k} N
0.4 m 30 y
x
454. Determine the magnitude of the moments of the force F about the x, y, and z axes. Solve the problem (a) using a Cartesian vector approach and (b) using a scalar approach.
z
A
4 ft y x 3 ft C
2 ft B F {4i 12j 3k} lb
229
455. Determine the moment of the force F about an axis extending between A and C. Express the result as a Cartesian vector.
z
A
4 ft y x 3 ft C
2 ft B F {4i 12j 3k} lb
230
*456. Determine the moment produced by force F about segment AB of the pipe assembly. Express the result as a Cartesian vector.
z C
F
{ 20i
10j
15k} N
4m A 3m x 4m B y
231
457. Determine the magnitude of the moment that the force F exerts about the y axis of the shaft. Solve the problem using a Cartesian vector approach and using a scalar approach.
z A 250 mm O 45 x
y
200 mm
B 50 mm 30
F
16 N
232
458. If F = 450 N, determine the magnitude of the moment produced by this force about the x axis.
z F 45 A B 60 60 100 mm x 300 mm 150 mm y
233
459. The friction at sleeve A can provide a maximum resisting moment of 125 N # m about the x axis. Determine the largest magnitude of force F that can be applied to the bracket so that the bracket will not turn.
A
z F 45 B 60 60 100 mm x 300 mm 150 mm y
234
*460. Determine the magnitude of the moment produced by the force of F = 200 N about the hinged axis (the x axis) of the door.
B
z 0.5 m
F 2m
200 N 15 2.5 m
y
A 1m x
235
461. If the tension in the cable is F = 140 lb, determine the magnitude of the moment produced by this force about the hinged axis, CD, of the panel.
z 4 ft 4 ft B 6 ft
F 6 ft A x C
D
6 ft y
236
462. Determine the magnitude of force F in cable AB in order to produce a moment of 500 lb # ft about the hinged axis CD, which is needed to hold the panel in the position shown.
z 4 ft 4 ft B 6 ft
F 6 ft A x C
D
6 ft y
237
463. The A-frame is being hoisted into an upright position by the vertical force of F = 80 lb. Determine the moment of this force about the y axis passing through points A and B when the frame is in the position shown.
z
F C A 3 ft x x 30 3 ft B y 15 6 ft y
238
*464. The A-frame is being hoisted into an upright position by the vertical force of F = 80 lb. Determine the moment of this force about the x axis when the frame is in the position shown.
z
F C A 3 ft x x 30 3 ft B y 15 6 ft y
239
465. The A-frame is being hoisted into an upright position by the vertical force of F = 80 lb. Determine the moment of this force about the y axis when the frame is in the position shown.
z
F C A 3 ft x x 30 3 ft B y 15 6 ft y
240
466. The flex-headed ratchet wrench is subjected to a force of P = 16 lb, applied perpendicular to the handle as shown. Determine the moment or torque this imparts along the vertical axis of the bolt at A.
60 10 in.
P
A
0.75 in.
467. If a torque or moment of 80 lb # in. is required to loosen the bolt at A, determine the force P that must be applied perpendicular to the handle of the flex-headed ratchet wrench.
60 10 in.
P
A
0.75 in.
241
*468. The pipe assembly is secured on the wall by the two brackets. If the flower pot has a weight of 50 lb, determine the magnitude of the moment produced by the weight about the OA axis.
z 4 ft A O 60
3 ft
4 ft 3 ft
B
x
30
y
242
469. The pipe assembly is secured on the wall by the two brackets. If the frictional force of both brackets can resist a maximum moment of 150 lb # ft, determine the largest weight of the flower pot that can be supported by the assembly without causing it to rotate about the OA axis.
z 4 ft A O 60
3 ft
4 ft 3 ft
B
x
30
y
243
470. A vertical force of F = 60 N is applied to the handle of the pipe wrench. Determine the moment that this force exerts along the axis AB (x axis) of the pipe assembly. Both the wrench and pipe assembly ABC lie in the x- y plane. Suggestion: Use a scalar analysis.
500 mm B x 45
z
A y F 150 mm
200 mm C
244
471. Determine the magnitude of the vertical force F acting on the handle of the wrench so that this force produces a component of moment along the AB axis (x axis) of the pipe assembly of (MA)x = 5 -5i6 N # m. Both the pipe assembly ABC and the wrench lie in the x -y plane. Suggestion: Use a scalar analysis.
z
A y 500 mm B x 45 200 mm C F 150 mm
*472. The frictional effects of the air on the blades of the standing fan creates a couple moment of MO = 6 N # m on the blades. Determine the magnitude of the couple forces at the base of the fan so that the resultant couple moment on the fan is zero.
MO
F 0.15 m
F 0.15 m
245
473. Determine the required magnitude of the couple moments M2 and M3 so that the resultant couple moment is zero.
M2 45
M3
M1
300 N m
246
474. The caster wheel is subjected to the two couples. Determine the forces F that the bearings exert on the shaft so that the resultant couple moment on the caster is zero.
500 N F
A 40 mm B F 100 mm 45 mm
50 mm
500 N
247
475. If F = 200 lb, determine the resultant couple moment.
B 2 ft 150 lb 2 ft 30 2 ft
2 ft F
4 3 5
2 ft
30 150 lb
4 3
5
F
A
248
*476. Determine the required magnitude of force F if the resultant couple moment on the frame is 200 lb # ft, clockwise.
B 2 ft 150 lb 2 ft 30 2 ft
2 ft F
4 3 5
2 ft
30 150 lb
4 3
5
F
A
249
477. The floor causes a couple moment of MA = 40 N # m and MB = 30 N # m on the brushes of the polishing machine. Determine the magnitude of the couple forces that must be developed by the operator on the handles so that the resultant couple moment on the polisher is zero. What is the magnitude of these forces if the brush at B suddenly stops so that MB = 0?
F A 0.3 m MB B F MA
478. If u = 30, determine the magnitude of force F so that the resultant couple moment is 100 N # m, clockwise.
300 N 300 mm u
15 F
30 F 30 u 300 N 15
250
479. If F = 200 N, determine the required angle u so that the resultant couple moment is zero.
300 N 300 mm u
15 F
30 F 30 u 300 N 15
*480. Two couples act on the beam. Determine the magnitude of F so that the resultant couple moment is 450 lb # ft, counterclockwise. Where on the beam does the resultant couple moment act?
200 lb 30 1.5 ft 200 lb 2 ft 1.25 ft 30
F F
251
481. The cord passing over the two small pegs A and B of the square board is subjected to a tension of 100 N. Determine the required tension P acting on the cord that passes over pegs C and D so that the resultant couple produced by the two couples is 15 N # m acting clockwise. Take u = 15.
C
u
300 mm
B 30 100 N
P
300 mm
100 N 30 A 45 D P
u
482. The cord passing over the two small pegs A and B of the board is subjected to a tension of 100 N. Determine the minimum tension P and the orientation u of the cord passing over pegs C and D, so that the resultant couple moment produced by the two cords is 20 N # m, clockwise.
C
u
300 mm
B 30 100 N
P
300 mm
100 N 30 A 45 D P
u
483. A device called a rolamite is used in various ways to replace slipping motion with rolling motion. If the belt, which wraps between the rollers, is subjected to a tension of 15 N, determine the reactive forces N of the top and bottom plates on the rollers so that the resultant couple acting on the rollers is equal to zero.
25 mm B
N
T
15 N
A 25 mm
30 T 15 N
N
252
*484. Two couples act on the beam as shown. Determine the magnitude of F so that the resultant couple moment is 300 lb # ft counterclockwise. Where on the beam does the resultant couple act?
1.5 ft
200 lb
200 lb
485. Determine the resultant couple moment acting on the beam. Solve the problem two ways: (a) sum moments about point O; and (b) sum moments about point A.
8 kN 45 30 B 0.3 m
1.5 m 2 kN
1.8 m
O A 30 2 kN 45 8 kN
253
486. Two couples act on the cantilever beam. If F = 6 kN, determine the resultant couple moment.
3m 5 kN
4 3 5
3m F B 0.5 m 30
A
30 F
0.5 m
4 3 5
5 kN
254
487. Determine the required magnitude of force F, if the resultant couple moment on the beam is to be zero.
3m 5 kN
4 3 5
3m F B 0.5 m 30
A
30 F
0.5 m
4 3 5
5 kN
255
*488. Two couples act on the frame. If the resultant couple moment is to be zero, determine the distance d between the 40-lb couple forces.
40 lb 1 ft 30
y 3 ft 60 lb
4 5 3
4 ft
B
4 3
5
60 lb d
30 A 40 lb
2 ft x
489. Two couples act on the frame. If d = 4 ft, determine the resultant couple moment. Compute the result by resolving each force into x and y components and (a) finding the moment of each couple (Eq. 413) and (b) summing the moments of all the force components about point A.
1 ft
y 3 ft 60 lb
4 5 3
4 ft
40 lb 30
B
4 3
5
60 lb d
30 A 40 lb
2 ft x
490. Two couples act on the frame. If d = 4 ft, determine the resultant couple moment. Compute the result by resolving each force into x and y components and (a) finding the moment of each couple (Eq. 413) and (b) summing the moments of all the force components about point B.
1 ft
y 3 ft 60 lb
4 5 3
4 ft
40 lb 30
B
4 3
5
60 lb d
30 A 40 lb
2 ft x
256
491. If M1 = 500 N # m, M2 = 600 N # m, and M3 = 450 N # m, determine the magnitude and coordinate direction angles of the resultant couple moment.
z M3
M2 30 M1 x y
257
*492. Determine the required magnitude of couple moments M1, M2, and M3 so that the resultant couple moment is MR = 5 - 300i + 450j - 600k6 N # m.
z M3
M2 30 M1 x y
258
493. If F = 80 N, determine the magnitude and coordinate direction angles of the couple moment. The pipe assembly lies in the xy plane.
z
F
300 mm 300 mm F
x 200 mm
200 mm
300 mm
y
259
494. If the magnitude of the couple moment acting on the pipe assembly is 50 N # m, determine the magnitude of the couple forces applied to each wrench. The pipe assembly lies in the xy plane.
z
F
300 mm 300 mm F
x 200 mm
200 mm
300 mm
y
260
495. From load calculations it is determined that the wing is subjected to couple moments Mx = 17 kip # ft and My = 25 kip # ft. Determine the resultant couple moments created about the x and y axes. The axes all lie in the same horizontal plane.
y My y
Mx 25
x
x
*496. Express the moment of the couple acting on the frame in Cartesian vector form. The forces are applied perpendicular to the frame. What is the magnitude of the couple moment? Take F = 50 N.
F
z
O y 3m
30 1.5 m x F
261
497. In order to turn over the frame, a couple moment is applied as shown. If the component of this couple moment along the x axis is Mx = 5- 20i6 N # m, determine the magnitude F of the couple forces.
F
z
O y 3m
30 1.5 m x F
498. Determine the resultant couple moment of the two couples that act on the pipe assembly. The distance from A to B is d = 400 mm. Express the result as a Cartesian vector.
250 mm { 50i} N C
{35k} N B d 30 { 35k} N
z
350 mm x A {50i} N
y
262
499. Determine the distance d between A and B so that the resultant couple moment has a magnitude of MR = 20 N # m.
250 mm { 50i} N C
{35k} N B d 30 { 35k} N
z
350 mm x A {50i} N
y
263
*4100. If M1 = 180 lb # ft, M2 = 90 lb # ft, and M3 = 120 lb # ft, determine the magnitude and coordinate direction angles of the resultant couple moment.
1 ft
z 150 lb ft M3 45 45
2 ft x 2 ft M2
2 ft
y 3 ft M1
264
4101. Determine the magnitudes of couple moments M1, M2, and M3 so that the resultant couple moment is zero.
M3 1 ft 2 ft x 2 ft M2
z 150 lb ft
2 ft
45 45
y 3 ft M1
265
4102. If F1 = 100 lb and F2 = 200 lb, determine the magnitude and coordinate direction angles of the resultant couple moment.
2 ft 250 lb x
z 3 ft 4 ft
F2 250 lb
F1 y F1
F2
266
4103. Determine the magnitude of couple forces F1 and F2 so that the resultant couple moment acting on the block is zero.
2 ft 250 lb x
z 3 ft 4 ft
F2 250 lb
F1 y F1
F2
267
*4104. Replace the force system acting on the truss by a resultant force and couple moment at point C.
A
200 lb 2 ft
150 lb
100 lb 2 ft B
3 4 5
2 ft
2 ft
6 ft
500 lb
C
268
4105. Replace the force system acting on the beam by an equivalent force and couple moment at point A.
3 kN 2.5 kN 1.5 kN 30
5 4 3
A 2m 4m 2m
B
4106. Replace the force system acting on the beam by an equivalent force and couple moment at point B.
3 kN 2.5 kN 1.5 kN 30
5 4 3
A 2m 4m 2m
B
269
270
4107. Replace the two forces by an equivalent resultant force and couple moment at point O. Set F = 20 lb.
y
20 lb 30
5 4 3
F
6 in. 1.5 in. O 40 x 2 in.
271
*4108. Replace the two forces by an equivalent resultant force and couple moment at point O. Set F = 15 lb.
y
20 lb 30
5 4 3
F
6 in. 1.5 in. O 40 x 2 in.
272
4109. Replace the force system acting on the post by a resultant force and couple moment at point A.
500 N 1m 30
0.5 m B
3 5 4
0.2 m
250 N 1m 300 N 1m A
273
4110. Replace the force and couple moment system acting on the overhang beam by a resultant force and couple moment at point A.
A
30 kN 30 0.3 m
26 kN
12 13 5
45 kN m
0.3 m B 2m 2m
1m
1m
274
4111. Replace the force system by a resultant force and couple moment at point O.
5 3 4
500 N
750 N 1m
200 N
O
200 N
1.25 m
1.25 m
275
*4112. Replace the two forces acting on the grinder by a resultant force and couple moment at point O. Express the results in Cartesian vector form.
z F1 {10i 15j 40k} N y A F2 100 mm O 150 mm 40 mm 25 mm x B { 15i 20j 30k} N
250 mm
276
4113. Replace the two forces acting on the post by a resultant force and couple moment at point O. Express the results in Cartesian vector form.
C FD 6m 2m D 3m x 7 kN
z A FB 5 kN
8m O 6m B y
277
4114. The three forces act on the pipe assembly. If F1 = 50 N and F2 = 80 N, replace this force system by an equivalent resultant force and couple moment acting at O. Express the results in Cartesian vector form.
z
180 N 1.25 m O
y
F2 x 0.5 m F1 0.75 m
4115. Handle forces F1 and F2 are applied to the electric drill. Replace this force system by an equivalent resultant force and couple moment acting at point O. Express the results in Cartesian vector form.
F2
{2j
4k} N z
0.15 m
F1
{6i
3j
10k} N
0.25 m
0.3 m O x y
278
*4116. Replace the force system acting on the pipe assembly by a resultant force and couple moment at point O. Express the results in Cartesian vector form.
z F2 { 10i 25j 20k} lb
F1
{ 20i
10j
25k}lb 2 ft
O 1.5 ft 2 ft x 2 ft
x
279
4117. The slab is to be hoisted using the three slings shown. Replace the system of forces acting on slings by an equivalent force and couple moment at point O. The force F1 is vertical.
F2 5 kN 60
z
F3 45 60 O 45 30 x 6m
4 kN
F1
6 kN
2m 2m
y
4118. The weights of the various components of the truck are shown. Replace this system of forces by an equivalent resultant force and specify its location measured from B.
B
3500 lb
5500 lb 14 ft
A 6 ft 2 ft
1750 lb
3 ft
280
4119. The weights of the various components of the truck are shown. Replace this system of forces by an equivalent resultant force and specify its location measured from point A.
B
3500 lb
5500 lb 14 ft
A 6 ft 2 ft
1750 lb
3 ft
*4120. The system of parallel forces acts on the top of the Warren truss. Determine the equivalent resultant force of the system and specify its location measured from point A.
1 kN 500 N 1m A 1m 500 N 1m
2 kN
500 N 1m
281
4121. The system of four forces acts on the roof truss. Determine the equivalent resultant force and specify its location along AB, measured from point A.
200 lb 275 lb 4 ft 300 lb 4 ft 150 lb 4 ft 30 B
A 30
282
4122. Replace the force and couple system acting on the frame by an equivalent resultant force and specify where the resultant's line of action intersects member AB, measured from A.
5 4
A 2 ft
3
150 lb
4 ft
500 lb ft B 3 ft 30 50 lb
C
283
4123. Replace the force and couple system acting on the frame by an equivalent resultant force and specify where the resultant's line of action intersects member BC, measured from B.
5 4
A 2 ft
3
150 lb
4 ft
500 lb ft B 3 ft 30 50 lb
C
284
*4124. Replace the force and couple moment system acting on the overhang beam by a resultant force, and specify its location along AB measured from point A.
A
30 kN 30 0.3 m 45 kN m
26 kN
12 5 13
0.3 m B
2m
1m
1m
2m
285
4125. Replace the force system acting on the frame by an equivalent resultant force and specify where the resultant's line of action intersects member AB, measured from point A.
35 lb A 2 ft
30 4 ft
20 lb B
3 ft 25 lb
2 ft C
286
4126. Replace the force system acting on the frame by an equivalent resultant force and specify where the resultant's line of action intersects member BC, measured from point B.
35 lb A 2 ft
30 4 ft
20 lb B
3 ft 25 lb
2 ft C
287
4127. Replace the force system acting on the post by a resultant force, and specify where its line of action intersects the post AB measured from point A.
1m 500 N
0.5 m B
3 5 4
0.2 m 30
250 N
1m 300 N 1m A
288
*4128. Replace the force system acting on the post by a resultant force, and specify where its line of action intersects the post AB measured from point B.
1m 500 N
0.5 m B
3 5 4
0.2 m 30
250 N
1m 300 N 1m A
4129. The building slab is subjected to four parallel column loadings. Determine the equivalent resultant force and specify its location (x, y) on the slab. Take F1 = 30 kN, F2 = 40 kN.
z
20 kN
50 kN
F1
F2 3m 8m 2m 6m 4m y
x
289
4130. The building slab is subjected to four parallel column loadings. Determine the equivalent resultant force and specify its location (x, y) on the slab. Take F1 = 20 kN, F2 = 50 kN.
z
20 kN
50 kN
F1
F2 3m 8m 2m 6m 4m y
x
4131. The tube supports the four parallel forces. Determine the magnitudes of forces FC and FD acting at C and D so that the equivalent resultant force of the force system acts through the midpoint O of the tube.
FD 600 N D A 400 mm
z
FC 500 N C 200 mm 200 mm y
O 400 mm
x
z B
290
*4132. Three parallel bolting forces act on the circular plate. Determine the resultant force, and specify its location (x, z) on the plate. F = 200 lb, FB = 100 lb, and A FC = 400 lb.
z
C 1.5 ft 45 x 30 B A FB
FC
FA
y
291
4133. The three parallel bolting forces act on the circular plate. If the force at A has a magnitude of FA = 200 lb, determine the magnitudes of FB and FC so that the resultant force FR of the system has a line of action that coincides with the y axis. Hint: This requires Mx = 0 and Mz = 0.
1.5 ft
z
C
FC
45 x 30 B A FB FA y
4134. If FA = 40 kN and FB = 35 kN, determine the magnitude of the resultant force and specify the location of its point of application (x, y) on the slab.
0.75 m FB 2.5 m 2.5 m 0.75 m 0.75 m x 3m 3m
z 30 kN 90 kN 20 kN FA
y
0.75 m
292
4135. If the resultant force is required to act at the center of the slab, determine the magnitude of the column loadings FA and FB and the magnitude of the resultant force.
0.75 m FB 2.5 m 2.5 m 0.75 m 0.75 m x 3m 3m
z 30 kN 90 kN 20 kN FA
y
0.75 m
*4136. Replace the parallel force system acting on the plate by a resultant force and specify its location on the xz plane.
z 0.5 m 1m
2 kN 5 kN 1m
1m 3 kN 0.5 m x
y
293
4137. If FA = 7 kN and FB = 5 kN, represent the force system acting on the corbels by a resultant force, and specify its location on the xy plane.
z FB 150 mm 100 mm FA 650 mm x O 100 mm 600 mm 150 mm y 6 kN 750 mm 8kN 700 mm
294
4138. Determine the magnitudes of FA and FB so that the resultant force passes through point O of the column.
150 mm 100 mm 6 kN
z FB 750 mm FA 650 mm x O 100 mm 600 mm 150 mm y 8kN 700 mm
295
4139. Replace the force and couple moment system acting on the rectangular block by a wrench. Specify the magnitude of the force and couple moment of the wrench and where its line of action intersects the xy plane.
600 lb ft
z
4 ft
450 lb 2 ft y x 3 ft
600 lb
300 lb
296
*4140. Replace the three forces acting on the plate by a wrench. Specify the magnitude of the force and couple moment for the wrench and the point P(y, z) where its line of action intersects the plate.
z
12 ft B y P z 12 ft FB { 60j} lb
A x FA { 80k}lb
FC
C { 40i} lb
y
297
4141. Replace the three forces acting on the plate by a wrench. Specify the magnitude of the force and couple moment for the wrench and the point P(x, y) where its line of action intersects the plate.
z FA A
FB
{800k} N
{500i} N
B y P x y x 4m C FC {300j} N 6m
298
4142. Replace the distributed loading with an equivalent resultant force, and specify its location on the beam measured from point A.
15 kN/m 10 kN/m
A B 3m 3m 3m
299
4143. Replace the distributed loading with an equivalent resultant force, and specify its location on the beam measured from point A.
8 kN/m 4 kN/m A B 3m 3m
300
*4144. Replace the distributed loading by an equivalent resultant force and specify its location, measured from point A.
800 N/m
200 N/m A 2m 3m B
4145. Replace the distributed loading with an equivalent resultant force, and specify its location on the beam measured from point A.
A
w0
w0
B L 2 L 2
301
4146. The distribution of soil loading on the bottom of a building slab is shown. Replace this loading by an equivalent resultant force and specify its location, measured from point O.
O 50 lb/ft 100 lb/ft
12 ft
300 lb/ft 9 ft
4147. Determine the intensities w1 and w2 of the distributed loading acting on the bottom of the slab so that this loading has an equivalent resultant force that is equal but opposite to the resultant of the distributed loading acting on the top of the plate.
A w1
3 ft
6 ft 300 lb/ft
1.5 ft
B
w2
302
*4148. The bricks on top of the beam and the supports at the bottom create the distributed loading shown in the second figure. Determine the required intensity w and dimension d of the right support so that the resultant force and couple moment about point A of the system are both zero.
0.5 m 3m
d
200 N/m
A 75 N/m 0.5 m d 3m w
303
4149. The wind pressure acting on a triangular sign is uniform. Replace this loading by an equivalent resultant force and couple moment at point O.
150 Pa
z 1.2 m 0.1 m
1.2 m
1m
y
O x
304
4150. The beam is subjected to the distributed loading. Determine the length b of the uniform load and its position a on the beam such that the resultant force and couple moment acting on the beam are zero.
b 40 lb/ft a
60 lb/ft 10 ft 6 ft
4151. Currently eighty-five percent of all neck injuries are caused by rear-end car collisions. To alleviate this problem, an automobile seat restraint has been developed that provides additional pressure contact with the cranium. During dynamic tests the distribution of load on the cranium has been plotted and shown to be parabolic. Determine the equivalent resultant force and its location, measured from point A.
A
12 lb/ft w w 12(1 2x2) lb/ft
0.5 ft
B 18 lb/ft
x
305
*4152. Wind has blown sand over a platform such that the intensity of the load can be approximated by the function w = 10.5x32 N>m. Simplify this distributed loading to an equivalent resultant force and specify its magnitude and location measured from A.
w 500 N/m
w
(0.5x3) N/m
A
x
10 m
4153. Wet concrete exerts a pressure distribution along the wall of the form. Determine the resultant force of this distribution and specify the height h where the bracing strut should be placed so that it lies through the line of action of the resultant force. The wall has a width of 5 m.
4m
p
p
(4 z /2) kPa
1
h 8 kPa
z
306
4154. Replace the distributed loading with an equivalent resultant force, and specify its location on the beam measured from point A.
w 8 kN/m 1 (4 2 x)2 x B 4m
w
A
307
4155. Replace the loading by an equivalent resultant force and couple moment at point A.
50 lb/ft 50 lb/ft B 4 ft
6 ft
100 lb/ft 60 A
308
*4156. Replace the loading by an equivalent resultant force and couple moment acting at point B.
50 lb/ft 50 lb/ft B 4 ft
6 ft
100 lb/ft 60 A
309
4157. The lifting force along the wing of a jet aircraft consists of a uniform distribution along AB, and a semiparabolic distribution along BC with origin at B. Replace this loading by a single resultant force and specify its location measured from point A.
w w 2880 lb/ft (2880 5x2) lb/ft
A
B C 12 ft 24 ft
x
4158. The distributed load acts on the beam as shown. Determine the magnitude of the equivalent resultant force and specify where it acts, measured from point A.
w w ( 2x 2 4x 16) lb/ft
A B 4 ft
x
310
4159. The distributed load acts on the beam as shown. Determine the maximum intensity wmax. What is the magnitude of the equivalent resultant force? Specify where it acts, measured from point B.
w w ( 2x 2 4x 16) lb/ft
A B 4 ft
x
311
*4160. The distributed load acts on the beam as shown. Determine the magnitude of the equivalent resultant force and specify its location, measured from point A.
w
w
(
2 x2 15
17 x 15
4) lb/ft
4 lb/ft 2 lb/ft A B x
10 ft
312
4161. If the distribution of the ground reaction on the pipe per foot of length can be approximated as shown, determine the magnitude of the resultant force due to this loading.
2.5 ft u
25 lb/ft
w 50 lb/ft
25 (1
cos u) lb/ft
313
4162. The beam is subjected to the parabolic loading. Determine an equivalent force and couple system at point A.
w
400 lb/ft
w
(25 x2)lb/ft A x
O 4 ft
4163. Two couples act on the frame. If the resultant couple moment is to be zero, determine the distance d between the 100-lb couple forces.
3 ft
100 lb 30 150 lb d 3 ft B
5 4 3
A 4 ft 30 100 lb
5
3
150 lb
4
314
*4164. Determine the coordinate direction angles a, b , g of F, which is applied to the end of the pipe assembly, so that the moment of F about O is zero.
F
20 lb
z
O 6 in.
y
10 in.
8 in.
x
6 in.
4165. Determine the moment of the force F about point O. The force has coordinate direction angles of a = 60, b = 120, g = 45. Express the result as a Cartesian vector.
F
20 lb
z
O 6 in.
y
10 in.
8 in.
x
6 in.
315
4166. The snorkel boom lift is extended into the position shown. If the worker weighs 160 lb, determine the moment of this force about the connection at A.
2 ft
25 ft
50 A
4167. Determine the moment of the force FC about the door hinge at A. Express the result as a Cartesian vector.
2.5 m
z C 1.5 m FC 250 N a
30
A B 1m 0.5 m
a
x
y
*4168. Determine the magnitude of the moment of the force FC about the hinged axis aa of the door.
2.5 m
z C 1.5 m FC 250 N a
30
A B 1m 0.5 m
a
x
y
316
4169. Express the moment of the couple acting on the pipe assembly in Cartesian vector form. Solve the problem (a) using Eq. 413 and (b) summing the moment of each force about point O. Take F = 525k6 N.
300 mm
z
O
200 mm 150 mm B F y
x 400 mm
F 200 mm A
4170. If the couple moment acting on the pipe has a magnitude of 400 N # m, determine the magnitude F of the vertical force applied to each wrench.
O
z
300 mm 200 mm 150 mm B F y
x 400 mm
F 200 mm A
317
4171. Replace the force at A by an equivalent resultant force and couple moment at point P. Express the results in Cartesian vector form.
P 4 ft 10 ft 6 ft
z
F
120 lb 8 ft
y
6 ft x
8 ft
A
*4172. The horizontal 30-N force acts on the handle of the wrench. Determine the moment of this force about point O. Specify the coordinate direction angles a, b , g of the moment axis.
50 mm
z 30 N 45 45 10 mm O x y
200 mm B
318
4173. The horizontal 30-N force acts on the handle of the wrench. What is the magnitude of the moment of this force about the z axis?
z 30 N 45 45 10 mm O x y
200 mm B 50 mm
319
51. Draw the free-body diagram of the 50-kg paper roll which has a center of mass at G and rests on the smooth blade of the paper hauler. Explain the significance of each force acting on the diagram. (See Fig. 57b.)
B
35 mm G A 30
52. Draw the free-body diagram of member AB, which is supported by a roller at A and a pin at B. Explain the significance of each force on the diagram. (See Fig. 57b.)
A 8 ft 30
390 lb
13 12 5
800 lb ft
4 ft B
3 ft
320
53. Draw the free-body diagram of the dumpster D of the truck, which has a weight of 5000 lb and a center of gravity at G. It is supported by a pin at A and a pin-connected hydraulic cylinder BC (short link). Explain the significance of each force on the diagram. (See Fig. 57b.)
D 3m A
G 1m B 20
1.5 m
30
C
*54. Draw the free-body diagram of the beam which supports the 80-kg load and is supported by the pin at A and a cable which wraps around the pulley at D. Explain the significance of each force on the diagram. (See Fig. 57b.)
D
4 3
5
A
B
E C
2m
2m
1.5 m
321
55. Draw the free-body diagram of the truss that is supported by the cable AB and pin C. Explain the significance of each force acting on the diagram. (See Fig. 57b.)
B
30
A
2m C
3 kN 2m 2m
4 kN 2m
56. Draw the free-body diagram of the crane boom AB which has a weight of 650 lb and center of gravity at G. The boom is supported by a pin at A and cable BC. The load of 1250 lb is suspended from a cable attached at B. Explain the significance of each force acting on the diagram. (See Fig. 57b.)
C
12 ft B 18 ft
13 12
5
G
A
30
322
57. Draw the free-body diagram of the "spanner wrench" subjected to the 20-lb force. The support at A can be considered a pin, and the surface of contact at B is smooth. Explain the significance of each force on the diagram. (See Fig. 57b.)
20 lb A
1 in.
B
6 in.
*58. Draw the free-body diagram of member ABC which is supported by a smooth collar at A, roller at B, and short link CD. Explain the significance of each force acting on the diagram. (See Fig. 57b.)
A 45 4m
2.5 kN
C
D
60
3m 4 kN m B
6m
323
59. Draw the free-body diagram of the bar, which has a negligible thickness and smooth points of contact at A, B, and C. Explain the significance of each force on the diagram. (See Fig. 57b.)
3 in. 30 C B A 8 in. 5 in.
10 lb 30
324
510. Draw the free-body diagram of the winch, which consists of a drum of radius 4 in. It is pin-connected at its center C, and at its outer rim is a ratchet gear having a mean radius of 6 in. The pawl AB serves as a two-force member (short link) and prevents the drum from rotating. Explain the significance of each force on the diagram. (See Fig. 57b.)
B
3 in. A 2 in. 6 in. C
4 in. 500 lb
511. Determine the normal reactions at A and B in Prob. 51.
325
*512. Determine the tension in the cord and the horizontal and vertical components of reaction at support A of the beam in Prob. 54.
513. Determine the horizontal and vertical components of reaction at C and the tension in the cable AB for the truss in Prob. 55.
326
514. Determine the horizontal and vertical components of reaction at A and the tension in cable BC on the boom in Prob. 56.
515. Determine the horizontal and vertical components of reaction at A and the normal reaction at B on the spanner wrench in Prob. 57.
*516. Determine the normal reactions at A and B and the force in link CD acting on the member in Prob. 58.
327
517. Determine the normal reactions at the points of contact at A, B, and C of the bar in Prob. 59.
518. Determine the horizontal and vertical components of reaction at pin C and the force in the pawl of the winch in Prob. 510.
328
519. Compare the force exerted on the toe and heel of a 120-lb woman when she is wearing regular shoes and stiletto heels. Assume all her weight is placed on one foot and the reactions occur at points A and B as shown.
120 lb 120 lb
A 5.75 in. 1.25 in.
B
A
B
0.75 in. 3.75 in.
329
*520. The train car has a weight of 24 000 lb and a center of gravity at G. It is suspended from its front and rear on the track by six tires located at A, B, and C. Determine the normal reactions on these tires if the track is assumed to be a smooth surface and an equal portion of the load is supported at both the front and rear tires.
C
G
6 ft 4 ft B A 5 ft
330
521. Determine the horizontal and vertical components of reaction at the pin A and the tension developed in cable BC used to support the steel frame.
B
60 kN 1m 1m 1m
30 kN m
5 3
3m
4
C
A
331
522. The articulated crane boom has a weight of 125 lb and center of gravity at G. If it supports a load of 600 lb, determine the force acting at the pin A and the force in the hydraulic cylinder BC when the boom is in the position shown.
4 ft A B 8 ft 1 ft 40 C 1 ft G
332
523. The airstroke actuator at D is used to apply a force of F = 200 N on the member at B. Determine the horizontal and vertical components of reaction at the pin A and the force of the smooth shaft at C on the member.
15
C
600 mm
B A 60 D F
200 mm
600 mm
333
*524. The airstroke actuator at D is used to apply a force of F on the member at B. The normal reaction of the smooth shaft at C on the member is 300 N. Determine the magnitude of F and the horizontal and vertical components of reaction at pin A.
C
15 600 mm
B A 60 D F
200 mm
600 mm
334
525. The 300-lb electrical transformer with center of gravity at G is supported by a pin at A and a smooth pad at B. Determine the horizontal and vertical components of reaction at the pin A and the reaction of the pad B on the transformer.
1.5 ft
A 3 ft G B
335
526. A skeletal diagram of a hand holding a load is shown in the upper figure. If the load and the forearm have masses of 2 kg and 1.2 kg, respectively, and their centers of mass are located at G1 and G2, determine the force developed in the biceps CD and the horizontal and vertical components of reaction at the elbow joint B. The forearm supporting system can be modeled as the structural system shown in the lower figure.
D
G1 C B A G2 D
G1 A 100 mm G2 135 mm
C
75 B 65 mm
336
527. As an airplane's brakes are applied, the nose wheel exerts two forces on the end of the landing gear as shown. Determine the horizontal and vertical components of reaction at the pin C and the force in strut AB.
20
C 30
B 400 mm
A 600 mm 2 kN
6 kN
337
*528. The 1.4-Mg drainpipe is held in the tines of the fork lift. Determine the normal forces at A and B as functions of the blade angle u and plot the results of force (vertical axis) versus u (horizontal axis) for 0 ... u ... 90.
A
0.4 m
B
u
338
529. The mass of 700 kg is suspended from a trolley which moves along the crane rail from d = 1.7 m to d = 3.5 m. Determine the force along the pin-connected knee strut BC (short link) and the magnitude of force at pin A as a function of position d. Plot these results of F and F BC A (vertical axis) versus d (horizontal axis).
d A C 2m
B
1.5 m
339
530. If the force of F = 100 lb is applied to the handle of the bar bender, determine the horizontal and vertical components of reaction at pin A and the reaction of the roller B on the smooth bar.
C
40 in.
F 60
B 5 in.
A
340
531. If the force of the smooth roller at B on the bar bender is required to be 1.5 kip, determine the horizontal and vertical components of reaction at pin A and the required magnitude of force F applied to the handle.
C
40 in.
F 60
B 5 in.
A
341
*532. The jib crane is supported by a pin at C and rod AB. If the load has a mass of 2 Mg with its center of mass located at G, determine the horizontal and vertical components of reaction at the pin C and the force developed in rod AB on the crane when x = 5 m.
3.2 m
4m A
C
0.2 m
B
D x
G
342
533. The jib crane is supported by a pin at C and rod AB. The rod can withstand a maximum tension of 40 kN. If the load has a mass of 2 Mg, with its center of mass located at G, determine its maximum allowable distance x and the corresponding horizontal and vertical components of reaction at C.
4m A
3.2 m 0.2 m B
C
D x
G
343
534. Determine the horizontal and vertical components of reaction at the pin A and the normal force at the smooth peg B on the member.
0.4 m
0.4 m
C 30 F 600 N
B A 30
344
535. The framework is supported by the member AB which rests on the smooth floor. When loaded, the pressure distribution on AB is linear as shown. Determine the length d of member AB and the intensity w for this case.
4 ft
7 ft 800 lb
A
B
w d
345
*536. Outriggers A and B are used to stabilize the crane from overturning when lifting large loads. If the load to be lifted is 3 Mg, determine the maximum boom angle u so that the crane does not overturn. The crane has a mass of 5 Mg and center of mass at GC, whereas the boom has a mass of 0.6 Mg and center of mass at GB.
4.5 m
GB
5m GC u
2.8 m A 0.7 m 2.3 m B
346
537. The wooden plank resting between the buildings deflects slightly when it supports the 50-kg boy. This deflection causes a triangular distribution of load at its ends, having maximum intensities of wA and wB. Determine wA and wB, each measured in N>m, when the boy is standing 3 m from one end as shown. Neglect the mass of the plank.
A wB 3m 0.45 m 6m
B
wA
0.3 m
347
538. Spring CD remains in the horizontal position at all times due to the roller at D. If the spring is unstretched when u = 0 and the bracket achieves its equilibrium position when u = 30, determine the stiffness k of the spring and the horizontal and vertical components of reaction at pin A.
D k
C
0.45 m
0.6 m B u
F A
300 N
348
539. Spring CD remains in the horizontal position at all times due to the roller at D. If the spring is unstretched when u = 0 and the stiffness is k = 1.5 kN>m, determine the smallest angle u for equilibrium and the horizontal and vertical components of reaction at pin A.
D k
C
0.45 m
0.6 m B u
F A
300 N
349
*540. The platform assembly has a weight of 250 lb and center of gravity at G1. If it is intended to support a maximum load of 400 lb placed at point G2, determine the smallest counterweight W that should be placed at B in order to prevent the platform from tipping over.
G2
2 ft
6 ft
G1
8 ft B
C 6 ft 1 ft 1 ft
D
350
541. Determine the horizontal and vertical components of reaction at the pin A and the reaction of the smooth collar B on the rod.
C 300 lb 450 lb B D 4 ft 1 ft 2 ft 1 ft
A
30
351
542. Determine the support reactions of roller A and the smooth collar B on the rod. The collar is fixed to the rod AB, but is allowed to slide along rod CD.
1m C B
A 900 N 1m 45 600 N m D 45 2m
352
543. The uniform rod AB has a weight of 15 lb. Determine the force in the cable when the rod is in the position shown.
B
5 ft
30 A 10 C
T
*544. Determine the horizontal and vertical components of force at the pin A and the reaction at the rocker B of the curved beam.
500 N 200 N
10 15
2m A B
353
545. The floor crane and the driver have a total weight of 2500 lb with a center of gravity at G. If the crane is required to lift the 500-lb drum, determine the normal reaction on both the wheels at A and both the wheels at B when the boom is in the position shown.
12 ft 3 ft D C 6 ft G A 2.2 ft 1.4 ft 8.4 ft E B 30
F
354
546. The floor crane and the driver have a total weight of 2500 lb with a center of gravity at G. Determine the largest weight of the drum that can be lifted without causing the crane to overturn when its boom is in the position shown.
3 ft D C 6 ft G A 2.2 ft 1.4 ft 8.4 ft E 30
12 ft
F
B
355
547. The motor has a weight of 850 lb. Determine the force that each of the chains exerts on the supporting hooks at A, B, and C. Neglect the size of the hooks and the thickness of the beam.
A
850 lb 0.5 ft 1 ft 1.5 ft C B 10 30 10
*548. Determine the force P needed to pull the 50-kg roller over the smooth step. Take u = 60.
P
u 0.1 m 0.6 m A 20 B
356
549. Determine the magnitude and direction u of the minimum force P needed to pull the 50-kg roller over the smooth step.
u
P
0.1 m 0.6 m A 20 B
357
550. The winch cable on a tow truck is subjected to a force of T = 6 kN when the cable is directed at u = 60. Determine the magnitudes of the total brake frictional force F for the rear set of wheels B and the total normal forces at both front wheels A and both rear wheels B for equilibrium. The truck has a total mass of 4 Mg and mass center at G.
G 1.25 m A 2m B 2.5 m 1.5 m
u F T
3m
358
551. Determine the minimum cable force T and critical angle u which will cause the tow truck to start tipping, i.e., for the normal reaction at A to be zero. Assume that the truck is braked and will not slip at B. The truck has a total mass of 4 Mg and mass center at G.x
G 1.25 m A 2m B 2.5 m 1.5 m
u F T
3m
*552. Three uniform books, each having a weight W and length a, are stacked as shown. Determine the maximum distance d that the top book can extend out from the bottom one so the stack does not topple over.
a
d
359
553. Determine the angle u at which the link ABC is held in equilibrium if member BD moves 2 in. to the right. The springs are originally unstretched when u = 0. Each spring has the stiffness shown. The springs remain horizontal since they are attached to roller guides.
kCF F
100 lb/ft
C
u 6 in. D B 6 in. E kAE 500 lb/ft A F
360
554. The uniform rod AB has a weight of 15 lb and the spring is unstretched when u = 0. If u = 30, determine the stiffness k of the spring.
k
6 ft u 3 ft B
A
555. The horizontal beam is supported by springs at its ends. Each spring has a stiffness of k = 5 kN>m and is originally unstretched so that the beam is in the horizontal position. Determine the angle of tilt of the beam if a load of 800 N is applied at point C as shown.
800 N
C A B
1m 3m
361
*556. The horizontal beam is supported by springs at its ends. If the stiffness of the spring at A is kA = 5 kN>m, determine the required stiffness of the spring at B so that if the beam is loaded with the 800 N it remains in the horizontal position. The springs are originally constructed so that the beam is in the horizontal position when it is unloaded.
800 N
C A B
1m 3m
362
557. The smooth disks D and E have a weight of 200 lb and 100 lb, respectively. If a horizontal force of P = 200 lb is applied to the center of disk E, determine the normal reactions at the points of contact with the ground at A, B, and C.
4 3
5
1.5 ft 1 ft D A B E C P
363
558. The smooth disks D and E have a weight of 200 lb and 100 lb, respectively. Determine the largest horizontal force P that can be applied to the center of disk E without causing the disk D to move up the incline.
4 3
5
1.5 ft 1 ft D A B E C P
364
559. A man stands out at the end of the diving board, which is supported by two springs A and B, each having a stiffness of k = 15 kN>m. In the position shown the board is horizontal. If the man has a mass of 40 kg, determine the angle of tilt which the board makes with the horizontal after he jumps off. Neglect the weight of the board and assume it is rigid.
A
1m B
3m
365
*560. The uniform rod has a length l and weight W. It is supported at one end A by a smooth wall and the other end by a cord of length s which is attached to the wall as shown. Show that for equilibrium it is required that h = [(s2 - l2)>3]1>2.
h
C
s A l B
366
561. If spring BC is unstretched with u = 0 and the bell crank achieves its equilibrium position when u = 15, determine the force F applied perpendicular to segment AD and the horizontal and vertical components of reaction at pin A. Spring BC remains in the horizontal postion at all times due to the roller at C.
C
k
2 kN/m B 150 300 mm A D 400 mm u F
367
562. The thin rod of length l is supported by the smooth tube. Determine the distance a needed for equilibrium if the applied load is P.
a A 2r B l
P
368
563. The cart supports the uniform crate having a mass of 85 kg. Determine the vertical reactions on the three casters at A, B, and C. The caster at B is not shown. Neglect the mass of the cart.
0.2 m
B A 0.5 m 0.6 m 0.4 m 0.2 m
0.1 m
C
0.35 m 0.35 m
369
*564. The pole for a power line is subjected to the two cable forces of 60 lb, each force lying in a plane parallel to the x- y plane. If the tension in the guy wire AB is 80 lb, determine the x, y, z components of reaction at the fixed base of the pole, O.
z
60 lb 1 ft A 45 45 4 ft 60 lb 80 lb
10 ft B 3 ft O y x
370
565. If P = 6 kN, x = 0.75 m and y = 1 m, determine the tension developed in cables AB, CD, and EF. Neglect the weight of the plate.
F
z
B P x E x 2m 2m C y A y
D
371
566. Determine the location x and y of the point of application of force P so that the tension developed in cables AB, CD, and EF is the same. Neglect the weight of the plate.
F
z
B P x E x 2m 2m C y A y
D
372
567. Due to an unequal distribution of fuel in the wing tanks, the centers of gravity for the airplane fuselage A and wings B and C are located as shown. If these components have weights W = 45 000 lb, W = 8000 lb, A B and W = 6000 lb, determine the normal reactions of the C wheels D, E, and F on the ground.
z
D
B
A C E
8 ft x
F 6 ft 8 ft 20 ft 6 ft
4 ft 3 ft
y
373
*568. Determine the magnitude of force F that must be exerted on the handle at C to hold the 75-kg crate in the position shown. Also, determine the components of reaction at the thrust bearing A and smooth journal bearing B.
z
0.1 m
A B
x
0.6 m 0.5 m 0.2 m 0.1 m F C y
374
569. The shaft is supported by three smooth journal bearings at A, B, and C. Determine the components of reaction at these bearings.
z 900 N 600 N A 0.6 m x 0.9 m 500 N 0.9 m B 0.9 m 450 N 0.6 m 0.9 m y C 0.9 m
375
570. Determine the tension in cables BD and CD and the x, y, z components of reaction at the ball-and-socket joint at A.
D
z
3m 300 N B 1.5 m A
x
0.5 m C 1m
y
376
571. The rod assembly is used to support the 250-lb cylinder. Determine the components of reaction at the ball-andsocket joint A, the smooth journal bearing E, and the force developed along rod CD. The connections at C and D are ball-and-socket joints.
z
D C A 1 ft 1 ft 1 ft 1 ft E 1.5 ft y
x
F
377
378
*572. Determine the components of reaction acting at the smooth journal bearings A, B, and C.
z 450 N C 0.6 m A 300 N m 45 0.4 m x 0.8 m 0.4 m B y
379
573. Determine the force components acting on the balland-socket at A, the reaction at the roller B and the tension on the cord CD needed for equilibrium of the quarter circular plate.
z 350 N 2m 1m A x 60 D 200 N C 200 N 3m B y
380
574. If the load has a weight of 200 lb, determine the x, y, z components of reaction at the ball-and-socket joint A and the tension in each of the wires.
2 ft 2 ft D
z
4 ft
F 3 ft A
B x 2 ft 2 ft
G 2 ft C
E
y
381
382
575. If the cable can be subjected to a maximum tension of 300 lb, determine the maximum force F which may be applied to the plate. Compute the x, y, z components of reaction at the hinge A for this loading.
C
z
3 ft F
2 ft
1 ft A 3 ft
y x B 9 ft
*576. The member is supported by a pin at A and a cable BC. If the load at D is 300 lb, determine the x, y, z components of reaction at the pin A and the tension in cable B C.
z 1 ft C
2 ft
A 2 ft B
x 6 ft 2 ft
2 ft
y D
383
577. The plate has a weight of W with center of gravity at G. Determine the distance d along line GH where the vertical force P = 0.75W will cause the tension in wire CD to become zero.
L 2 H x L 2
z B P L 2 D F A G d L 2 E C y
384
578. The plate has a weight of W with center of gravity at G. Determine the tension developed in wires AB, CD, and EF if the force P = 0.75W is applied at d = L/2.
P L 2
z B D F A G d L 2 E C y
L 2 H x L 2
385
579. The boom is supported by a ball-and-socket joint at A and a guy wire at B. If the 5-kN loads lie in a plane which is parallel to the xy plane, determine the x, y, z components of reaction at A and the tension in the cable at B.
z 5 kN 30
5 kN
30 3m
2m A x 1.5 m B y
386
*580. The circular door has a weight of 55 lb and a center of gravity at G. Determine the x, y, z components of reaction at the hinge A and the force acting along strut CB needed to hold the door in equilibrium. Set u = 45.
z
A G x 3 ft u y B 3 ft
C
387
581. The circular door has a weight of 55 lb and a center of gravity at G. Determine the x, y, z components of reaction at the hinge A and the force acting along strut CB needed to hold the door in equilibrium. Set u = 90.
z
A G x 3 ft u y B 3 ft
C
388
582. Member AB is supported at B by a cable and at A by a smooth fixed square rod which fits loosely through the square hole of the collar. If F = 520i - 40j - 75k6 lb, determine the x, y, z components of reaction at A and the tension in the cable.
z 8 ft C
6 ft y
A
12 ft B
4 ft
F
x
389
583. Member AB is supported at B by a cable and at A by a smooth fixed square rod which fits loosely through the square hole of the collar. Determine the tension in cable BC if the force F = 5 - 45k6 lb.
z 8 ft C
6 ft y
A
12 ft B
4 ft
F
x
390
*584. Determine the largest weight of the oil drum that the floor crane can support without overturning. Also, what are the vertical reactions at the smooth wheels A, B, and C for this case. The floor crane has a weight of 300 lb, with its center of gravity located at G.
z 10 ft
30
3 ft 1.5ft C
G
A 2.5 ft 2.5 ft x B 4 ft
2 ft
1 ft y
391
585. The circular plate has a weight W and center of gravity at its center. If it is supported by three vertical cords tied to its edge, determine the largest distance d from the center to where any vertical force P can be applied so as not to cause the force in any one of the cables to become zero.
B
120 120 r A 120 d
P
C
392
586. Solve Prob. 585 if the plate's weight W is neglected.
B
120 120 r A 120 d
P
C
393
587. A uniform square table having a weight W and sides a is supported by three vertical legs. Determine the smallest vertical force P that can be applied to its top that will cause it to tip over.
a/2 a/2
a
394
*588. Determine the horizontal and vertical components of reaction at the pin A and the force in the cable BC. Neglect the thickness of the members.
C 30 B 200 N/m 3m
4m
100 N
A 4.5 m
395
589. Determine the horizontal and vertical components of reaction at the pin A and the reaction at the roller B required to support the truss. Set F = 600 N.
A
2m 45 2m F F 2m F 2m B
590. If the roller at B can sustain a maximum load of 3 kN, determine the largest magnitude of each of the three forces F that can be supported by the truss.
A
2m 45 2m F F 2m F 2m B
396
591. Determine the normal reaction at the roller A and horizontal and vertical components at pin B for equilibrium of the member.
A
10 kN 0.6 m 0.6 m
0.8 m
6 kN
60 0.4 m B
397
*592. The shaft assembly is supported by two smooth journal bearings A and B and a short link DC. If a couple moment is applied to the shaft as shown, determine the components of force reaction at the journal bearings and the force in the link. The link lies in a plane parallel to the yz plane and the bearings are properly aligned on the shaft.
z D 30 20 120 mm C B
250 mm A x 250 N m 400 mm
300 mm y
398
593. Determine the reactions at the supports A and B of the frame.
5 kip 8 ft
7 kip 6 ft
10 kip 2 kip 6 ft
A
8 ft
0.5 kip 6 ft B
399
594. A skeletal diagram of the lower leg is shown in the lower figure. Here it can be noted that this portion of the leg is lifted by the quadriceps muscle attached to the hip at A and to the patella bone at B. This bone slides freely over cartilage at the knee joint. The quadriceps is further extended and attached to the tibia at C. Using the mechanical system shown in the upper figure to model the lower leg, determine the tension in the quadriceps at C and the magnitude of the resultant force at the femur (pin), D, in order to hold the lower leg in the position shown. The lower leg has a mass of 3.2 kg and a mass center at G1; the foot has a mass of 1.6 kg and a mass center at G2.
75 mm 25 mm A B
C
350 mm 300 mm
D 75 G1 G2
A
B D
C
400
595. A vertical force of 80 lb acts on the crankshaft. Determine the horizontal equilibrium force P that must be applied to the handle and the x, y, z components of force at the smooth journal bearing A and the thrust bearing B. The bearings are properly aligned and exert only force reactions on the shaft.
A
z
80 lb B
10 in.
y
14 in. 14 in. 6 in. x 8 in. 4 in. P
401
*596. The symmetrical shelf is subjected to a uniform load of 4 kPa. Support is provided by a bolt (or pin) located at each end A and A and by the symmetrical brace arms, which bear against the smooth wall on both sides at B and B . Determine the force resisted by each bolt at the wall and the normal force at B for equilibrium.
A
4 kPa
B
A 0.15 m B 0.2 m
1.5 m
402
61. Determine the force in each member of the truss, and state if the members are in tension or compression.
600 N
D
2m 900 N E C
2m A 2m B
403
62. The truss, used to support a balcony, is subjected to the loading shown. Approximate each joint as a pin and determine the force in each member. State whether the members are in tension or compression. Set P1 = 600 lb, P2 = 400 lb.
P1
P2 B 45 45
A
C
4 ft
E D 4 ft 4 ft
404
63. The truss, used to support a balcony, is subjected to the loading shown. Approximate each joint as a pin and determine the force in each member. State whether the members are in tension or compression. Set P1 = 800 lb, P2 = 0.
P1
P2 B 45 45
A
C
4 ft
E D 4 ft 4 ft
405
*64. Determine the force in each member of the truss and state if the members are in tension or compression. Assume each joint as a pin. Set P = 4 kN.
A
2P P B C P
4m E
D 4m 4m
406
65. Assume that each member of the truss is made of steel having a mass per length of 4 kg/m. Set P = 0, determine the force in each member, and indicate if the members are in tension or compression. Neglect the weight of the gusset plates and assume each joint is a pin. Solve the problem by assuming the weight of each member can be represented as a vertical force, half of which is applied at the end of each member.
2P P B A C P
4m E
D 4m 4m
407
66. Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = 2 kN and P2 = 1.5 kN.
A
B
30 E 3m P1 D
30 C 3m P2
408
67. Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = P2 = 4 kN.
A
B
30 E 3m P1 D
30 C 3m P2
409
*68. Determine the force in each member of the truss, and state if the members are in tension or compression. Set P = 800 lb.
500 lb 3 ft F 3 ft E 3 ft D 3 ft A B P
C
410
69. Remove the 500-lb force and then determine the greatest force P that can be applied to the truss so that none of the members are subjected to a force exceeding either 800 lb in tension or 600 lb in compression.
500 lb 3 ft F 3 ft E 3 ft D 3 ft A B P
C
411
610. Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = 800 lb, P2 = 0.
4 ft
4 ft P1 C B
4 ft
4 ft
P2
D E G F
6 ft
A
412
611. Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = 600 lb, P2 = 400 lb.
4 ft
4 ft P1 C B
4 ft
4 ft
P2
D E G F
6 ft
A
413
*612. Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = 240 lb, P2 = 100 lb.
5 ft
C
D
P1
P2
B 12 ft
A
414
613. Determine the largest load P2 that can be applied to the truss so that the force in any member does not exceed 500 lb (T) or 350 lb (C). Take P1 = 0.
5 ft
C
D
P1
P2
B 12 ft
A
415
614. Determine the force in each member of the truss, and state if the members are in tension or compression. Set P = 2500 lb.
1200 lb 4 ft E 4 ft F 30 4 ft
P 4 ft D
1200 lb 4 ft C
G 30 B
A
416
615. Remove the 1200-lb forces and determine the greatest force P that can be applied to the truss so that none of the members are subjected to a force exceeding either 2000 lb in tension or 1500 lb in compression.
1200 lb 4 ft E 4 ft F 30 4 ft
P 4 ft D
1200 lb 4 ft C
G 30 B
A
417
*616. Determine the force in each member of the truss, and state if the members are in tension or compression. Set P = 5 kN.
D
E 1.5 m
1.5 m A C
1.5 m P 2m B 2m
418
617. Determine the greatest force P that can be applied to the truss so that none of the members are subjected to a force exceeding either 2.5 kN in tension or 2 kN in compression.
D
E 1.5 m
1.5 m A C
1.5 m P 2m B 2m
419
618. Determine the force in each member of the truss, and state if the members are in tension or compression.
900 lb 4 ft F 3 ft E 4 ft
600 lb 4 ft D
C A B
420
619. The truss is fabricated using members having a weight of 10 lb>ft. Remove the external forces from the truss, and determine the force in each member due to the weight of the members. State whether the members are in tension or compression. Assume that the total force acting on a joint is the sum of half of the weight of every member connected to the joint.
900 lb 4 ft F 3 ft E 4 ft
600 lb 4 ft D
C A B
421
422
*620. Determine the force in each member of the truss and state if the members are in tension or compression. The load has a mass of 40 kg.
D E 3.5 m F C G B 2.5 m
0.1 m
A 6m
423
621. Determine the largest mass m of the suspended block so that the force in any member does not exceed 30 kN (T) or 25 kN (C).
D E 3.5 m F C G B 2.5 m
0.1 m
A 6m
424
622. Determine the force in each member of the truss, and state if the members are in tension or compression.
400 N E
600 N
D
A
45
45 B 2m
45
45
C
2m
425
623. The truss is fabricated using uniform members having a mass of 5 kg>m. Remove the external forces from the truss, and determine the force in each member due to the weight of the truss. State whether the members are in tension or compression. Assume that the total force acting on a joint is the sum of half of the weight of every member connected to the joint.
400 N E
600 N
D
A
45
45 B 2m
45
45
C
2m
426
*624. Determine the force in each member of the truss, and state if the members are in tension or compression. Set P = 4 kN.
A P
3m F
3m E
3m
3m B C D P
427
625. Determine the greatest force P that can be applied to the truss so that none of the members are subjected to a force exceeding either 1.5 kN in tension or 1 kN in compression.
A P
3m F
3m E
3m
3m B C D P
428
626. A sign is subjected to a wind loading that exerts horizontal forces of 300 lb on joints B and C of one of the side supporting trusses. Determine the force in each member of the truss and state if the members are in tension or compression.
300 lb
C
12 ft 13 ft 5 ft 300 lb B 13 ft 45 A E 12 ft D
429
627. Determine the force in each member of the double scissors truss in terms of the load P and state if the members are in tension or compression.
B
C
L/3 A L/3 P E L/3 P F L/3 D
430
*628. Determine the force in each member of the truss in terms of the load P, and indicate whether the members are in tension or compression.
P
B
d A F D d C
E d d/2 d/2 d
431
629. If the maximum force that any member can support is 4 kN in tension and 3 kN in compression, determine the maximum force P that can be applied at joint B. Take d = 1 m.
A
P
B
d C F D d
E d d/2 d/2 d
432
630. The two-member truss is subjected to the force of 300 lb. Determine the range of u for application of the load so that the force in either member does not exceed 400 lb (T) or 200 lb (C).
B
3 ft
A u 4 ft 300 lb
C
433
434
631. The internal drag truss for the wing of a light airplane is subjected to the forces shown. Determine the force in members BC, BH, and HC, and state if the members are in tension or compression.
J
I
H
G
F 2 ft E
A 2 ft
B 2 ft 80 lb
C 2 ft
D
60 lb 80 lb
1.5 ft 40 lb
435
*632. The Howe bridge truss is subjected to the loading shown. Determine the force in members HD, CD, and GD, and state if the members are in tension or compression.
30 kN 20 kN J I 20 kN H
40 kN
G
F 4m
A B C 16 m, 4@4m D
E
436
633. The Howe bridge truss is subjected to the loading shown. Determine the force in members HI, HB, and BC, and state if the members are in tension or compression.
30 kN 20 kN J I 20 kN H
40 kN
G
F 4m
A B C 16 m, 4@4m D
E
437
634. Determine the force in members JK, CJ, and CD of the truss, and state if the members are in tension or compression.
K 3m A B 2m 2m C 2m L
J
I H G
D 2m
E 2m
F 2m
4 kN
5 kN
8 kN
6 kN
438
635. Determine the force in members HI, FI, and EF of the truss, and state if the members are in tension or compression.
K 3m A B 2m 2m C 2m L
J
I H G
D 2m
E 2m
F 2m
4 kN
5 kN
8 kN
6 kN
439
*636. Determine the force in members BC, CG, and GF of the Warren truss. Indicate if the members are in tension or compression.
3m A
3m B C
3m D
3m E G 3m 6 kN 3m 8 kN F 3m
637. Determine the force in members CD, CF, and FG of the Warren truss. Indicate if the members are in tension or compression.
3m A
3m B C
3m D
3m E G 3m 6 kN 3m 8 kN F 3m
440
638. Determine the force in members DC, HC, and HI of the truss, and state if the members are in tension or compression.
40 kN 2m E 2m
50 kN 2m D 1.5 m
F G H I
C
30 kN 1.5 m 40 kN 1.5 m
B
A
441
639. Determine the force in members ED, EH, and GH of the truss, and state if the members are in tension or compression.
40 kN 2m E 2m
50 kN 2m D 1.5 m
F G H I
C
30 kN 1.5 m 40 kN 1.5 m
B
A
442
*640. Determine the force in members GF, GD, and CD of the truss and state if the members are in tension or compression.
C B 4 ft 3 ft A H 4 ft 4 ft G 4 ft 3 ft F 4 ft E D 260 lb
13 5 12
641. Determine the force in members BG, BC, and HG of the truss and state if the members are in tension or compression.
C B 4 ft 3 ft A H 4 ft 4 ft G 4 ft 3 ft F 4 ft E D 260 lb
13 5 12
443
642. Determine the force in members IC and CG of the truss and state if these members are in tension or compression. Also, indicate all zero-force members.
B
C
D
2m I J
2m A H 1.5 m 1.5 m 6 kN G 1.5 m 6 kN F 1.5 m E
643. Determine the force in members JE and GF of the truss and state if these members are in tension or compression. Also, indicate all zero-force members.
B
C
D
2m I J
2m A H 1.5 m 1.5 m 6 kN G 1.5 m 6 kN F 1.5 m E
444
*644. Determine the force in members JI, EF, EI, and JE of the truss, and state if the members are in tension or compression.
8 ft 8 ft 8 ft N A 8 ft B
1500 lb 1000 lb 1000 lb 900 lb L M C D E F H G 8 ft 8 ft 8 ft 8 ft 8 ft K J I
445
645. Determine the force in members CD, LD, and KL of the truss, and state if the members are in tension or compression.
8 ft 8 ft 8 ft N A 8 ft B
1500 lb 1000 lb 1000 lb 900 lb L M C D E F H G 8 ft 8 ft 8 ft 8 ft 8 ft K J I
446
646. Determine the force developed in members BC and CH of the roof truss and state if the members are in tension or compression.
C 2 kN 0.8 m A H 1m 2m G 1.5 kN 2m F 1m D 1.5 m
B
E
447
647. Determine the force in members CD and GF of the truss and state if the members are in tension or compression. Also indicate all zero-force members.
C 2 kN 0.8 m A H 1m 2m G 1.5 kN 2m F 1m D 1.5 m
B
E
448
*648. Determine the force in members IJ, EJ, and CD of the Howe truss, and state if the members are in tension or compression.
5 kN 3 kN 4m A B 2m L
6 kN 5 kN J K I H 2 kN G C 2m D 2m E 2m F 2m 2m 4 kN 4 kN
449
649. Determine the force in members KJ, KC, and BC of the Howe truss, and state if the members are in tension or compression.
5 kN 3 kN 4m A B 2m L
6 kN 5 kN J K I H 2 kN G C 2m D 2m E 2m F 2m 2m 4 kN 4 kN
450
650. Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = 20 kN, P2 = 10 kN.
B
C
D
2m A G 1.5 m P1 1.5 m 1.5 m P2 F 1.5 m E
451
452
651. Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = 40 kN, P2 = 20 kN.
B
C
D
2m A G 1.5 m P1 1.5 m 1.5 m P2 F 1.5 m E
453
454
*652. Determine the force in members KJ, NJ, ND, and CD of the K truss. Indicate if the members are in tension or compression. Hint: Use sections aa and bb.
L 15 ft M 15 ft A B 1200 lb
K N
a b
J
I O
H P G F
C
a b
D
E
1500 lb 20 ft 20 ft
1800 lb 20 ft 20 ft
20 ft
20 ft
455
653. Determine the force in members JI and DE of the K truss. Indicate if the members are in tension or compression.
L 15 ft M 15 ft A B 1200 lb
K N
a b
J
I O
H P G F
C
a b
D
E
1500 lb 20 ft 20 ft
1800 lb 20 ft 20 ft
20 ft
20 ft
456
654. The space truss supports a force F = 5 - 500i + 600j + 400k6 lb. Determine the force in each member, and state if the members are in tension or compression.
8 ft 6 ft D 6 ft
z
C
F
B A 6 ft 6 ft x y
457
458
655. The space truss supports a force F = 5600i + 450j - 750k6 lb. Determine the force in each member, and state if the members are in tension or compression.
8 ft 6 ft D 6 ft
z
C
F
B A 6 ft 6 ft x y
459
460
*656. Determine the force in each member of the space truss and state if the members are in tension or compression. The truss is supported by ball-and-socket joints at A, B, and E. Set F = 5800j6 N. Hint: The support reaction at E acts along member EC. Why?
5m
z
D F
1m
A
2m C y
E 2m B 1.5 m
x
461
657. Determine the force in each member of the space truss and state if the members are in tension or compression. The truss is supported by ball-and-socket joints at A, B, and E. Set F = 5- 200i + 400j6 N. Hint: The support reaction at E acts along member EC. Why?
5m
z
D F
1m
A
2m C y
E 2m B 1.5 m
x
462
658. Determine the force in members BE, DF, and BC of the space truss and state if the members are in tension or compression.
2m
E D 2m C 3m 2m B { 2k} kN { 2k} kN
2m F A
463
659. Determine the force in members AB, CD, ED, and CF of the space truss and state if the members are in tension or compression.
2m
E D 2m C 3m 2m B { 2k} kN { 2k} kN
2m F A
464
*660. Determine the force in the members AB, AE, BC, BF, BD, and BE of the space truss, and state if the members are in tension or compression.
E
z
F 4 ft D C x 2 ft 600 lb 4 ft 2 ft 300 lb 400 lb A B 4 ft y
465
466
661. Determine the force in the members EF, DF, CF, and CD of the space truss, and state if the members are in tension or compression.
E
z
F 4 ft D C x 2 ft 600 lb 4 ft 2 ft 300 lb 400 lb A B 4 ft y
467
468
662. If the truss supports a force of F = 200 N, determine the force in each member and state if the members are in tension or compression.
z 200 mm 200 mm
D C 200 mm x y 500 mm A 300 mm E B 200 mm
F
469
663. If each member of the space truss can support a maximum force of 600 N in compression and 800 N in tension, determine the greatest force F the truss can support.
z 200 mm 200 mm
D C 200 mm x y 500 mm A 300 mm E B 200 mm
F
470
*664. Determine the force developed in each member of the space truss and state if the members are in tension or compression. The crate has a weight of 150 lb.
z 6 ft C
D 6 ft 6 ft
B A 6 ft y x
471
665. Determine the force in members FE and ED of the space truss and state if the members are in tension or compression. The truss is supported by a ball-and-socket joint at C and short links at A and B.
z { 500k} lb G {200j} lb
F D E
6 ft
C A B 3 ft x 3 ft 2 ft y 6 ft 4 ft
472
666. Determine the force in members GD, GE, and FD of the space truss and state if the members are in tension or compression.
G
z { 500k} lb {200j} lb
F D E
6 ft
C A B 3 ft x 3 ft 2 ft y 6 ft 4 ft
473
667. Determine the force P required to hold the 100-lb weight in equilibrium.
D
C
B A P
474
*668. Determine the force P required to hold the 150-kg crate in equilibrium.
B
A
C
P
475
669. Determine the force P required to hold the 50-kg mass in equilibrium.
B
C
A
P
476
670. Determine the force P needed to hold the 20-lb block in equilibrium.
B
C
A P
477
671. Determine the force P needed to support the 100-lb weight. Each pulley has a weight of 10 lb. Also, what are the cord reactions at A and B?
A
C 2 in.
2 in. B
2 in. P
478
*672. The cable and pulleys are used to lift the 600-lb stone. Determine the force that must be exerted on the cable at A and the corresponding magnitude of the resultant force the pulley at C exerts on pin B when the cables are in the position shown.
A
B C 30
D P
479
673. If the peg at B is smooth, determine the components of reaction at the pin A and fixed support C.
500 N 600 mm 600 mm
800 mm
B 45
C
A 900 N m
480
674. Determine the horizontal and vertical components of reaction at pins A and C.
A 3 ft
150 lb 100 lb
B 2 ft 2 ft 45
C
481
675. The compound beam is fixed at A and supported by rockers at B and C. There are hinges (pins) at D and E. Determine the components of reaction at the supports.
A D
15 kN 30 kN m C 6m 2m 2m 2m 6m
B
E
482
*676. The compound beam is pin-supported at C and supported by rollers at A and B. There is a hinge (pin) at D. Determine the components of reaction at the supports. Neglect the thickness of the beam.
8 kip 15 kip ft B
12 kip
5 4 3
A
D
C
30 4 kip
6 ft
4 ft 2 ft
8 ft
8 ft
8 ft
483
677. The compound beam is supported by a rocker at B and is fixed to the wall at A. If it is hinged (pinned) together at C, determine the components of reaction at the supports. Neglect the thickness of the beam.
A 4 ft
500 lb 200 lb
13 5 12
60 C
4000 lb ft B
4 ft
8 ft
4 ft
484
678. Determine the horizontal and vertical components of reaction at pins A and C of the two-member frame.
200 N/ m
B A
3m
C 3m
485
679. If a force of F = 50 N acts on the rope, determine the cutting force on the smooth tree limb at D and the horizontal and vertical components of force acting on pin A. The rope passes through a small pulley at C and a smooth ring at E.
30 mm D
100 mm B C A
E
F
50 N
486
*680. Two beams are connected together by the short link BC. Determine the components of reaction at the fixed support A and at pin D.
12 kN
10 kN
C A 1m 3m B 1.5 m D 1.5 m
487
681. The bridge frame consists of three segments which can be considered pinned at A, D, and E, rocker supported at C and F, and roller supported at B. Determine the horizontal and vertical components of reaction at all these supports due to the loading shown.
2 kip/ft A 15 ft 20 ft B 30 ft D 15 ft E
C 5 ft
F 5 ft
488
682. If the 300-kg drum has a center of mass at point G, determine the horizontal and vertical components of force acting at pin A and the reactions on the smooth pads C and D. The grip at B on member DAB resists both horizontal and vertical components of force at the rim of the drum.
P
600 mm E A 60 mm 60 mm 390 mm 100 mm D G C 30 B
489
683. Determine the horizontal and vertical components of reaction that pins A and C exert on the two-member arch.
2 kN B
1.5 kN 1.5 m 1m A 0.5 m C
490
*684. The truck and the tanker have weights of 8000 lb and 20 000 lb respectively. Their respective centers of gravity are located at points G1 and G2. If the truck is at rest, determine the reactions on both wheels at A, at B, and at C. The tanker is connected to the truck at the turntable D which acts as a pin.
G2
D
G1
A 15 ft 10 ft
B 9 ft 5 ft
C
491
685. The platform scale consists of a combination of third and first class levers so that the load on one lever becomes the effort that moves the next lever. Through this arrangement, a small weight can balance a massive object. If x = 450 mm, determine the required mass of the counterweight S required to balance a 90-kg load, L.
100 mm
250 mm
150 mm H F
E
G
C
D 350 mm B x L S
150 mm A
492
686. The platform scale consists of a combination of third and first class levers so that the load on one lever becomes the effort that moves the next lever. Through this arrangement, a small weight can balance a massive object. If x = 450 mm and, the mass of the counterweight S is 2 kg, determine the mass of the load L required to maintain the balance.
100 mm
250 mm
150 mm H F
E
G
C
D 350 mm B x L S
150 mm A
493
687. The hoist supports the 125-kg engine. Determine the force the load creates in member DB and in member FB, which contains the hydraulic cylinder H.
1m G F
2m E
2m
H D 1m C A B
2m
1m
494
*688. The frame is used to support the 100-kg cylinder E. Determine the horizontal and vertical components of reaction at A and D.
D 0.6 m
1.2 m r C 0.1 m
A E
495
689. Determine the horizontal and vertical components of reaction which the pins exert on member AB of the frame.
A
300 lb 60
B
4 ft
E D 3 ft 3 ft 500 lb C
496
690. Determine the horizontal and vertical components of reaction which the pins exert on member EDC of the frame.
A
300 lb 60
B
4 ft
E D 3 ft 3 ft 500 lb C
497
691. The clamping hooks are used to lift the uniform smooth 500-kg plate. Determine the resultant compressive force that the hook exerts on the plate at A and B, and the pin reaction at C.
P P 80 mm 150 mm P
C A B
498
*692. The wall crane supports a load of 700 lb. Determine the horizontal and vertical components of reaction at the pins A and D. Also, what is the force in the cable at the winch W?
4 ft
D
4 ft
4 ft C
A
B
E 60 W 700 lb
499
693. The wall crane supports a load of 700 lb. Determine the horizontal and vertical components of reaction at the pins A and D. Also, what is the force in the cable at the winch W? The jib ABC has a weight of 100 lb and member BD has a weight of 40 lb. Each member is uniform and has a center of gravity at its center.
D
4 ft 4 ft 4 ft C A B
E 60 W 700 lb
500
694. The lever-actuated scale consists of a series of compound levers. If a load of weight W = 150 lb is placed on the platform, determine the required weight of the counterweight S to balance the load. Is it necessary to place the load symmetrically on the platform? Explain.
S
1.25 in. 4 in. M J K
L
W 1.5 in. F G E A 7.5 in. H D 7.5 in. 4.5 in. I 1.5 in. C B
501
695. If P = 75 N, determine the force F that the toggle clamp exerts on the wooden block.
140 mm
85 mm P 140 mm
50 mm 50 mm
A D C B
20 mm E F P
502
*696. If the wooden block exerts a force of F = 600 N on the toggle clamp, determine the force P applied to the handle.
140 mm
85 mm P 140 mm
50 mm 50 mm
A D C B
20 mm E F P
503
697. The pipe cutter is clamped around the pipe P. If the wheel at A exerts a normal force of FA = 80 N on the pipe, determine the normal forces of wheels B and C on the pipe. The three wheels each have a radius of 7 mm and the pipe has an outer radius of 10 mm.
A P
C 10 mm B 10 mm
504
698. A 300-kg counterweight, with center of mass at G, is mounted on the pitman crank AB of the oil-pumping unit. If a force of F = 5 kN is to be developed in the fixed cable attached to the end of the walking beam DEF, determine the torque M that must be supplied by the motor.
1.75 m
2.50 m
D 30 B G 30 0.5 m 0.65 m
E
F
M A
F
505
699. A 300-kg counterweight, with center of mass at G, is mounted on the pitman crank AB of the oil-pumping unit. If the motor supplies a torque of M = 2500 N # m, determine the force F developed in the fixed cable attached to the end of the walking beam DEF.
1.75 m
2.50 m
D 30 B G 30 0.5 m 0.65 m
E
F
M A
F
506
*6100. The two-member structure is connected at C by a pin, which is fixed to BDE and passes through the smooth slot in member AC. Determine the horizontal and vertical components of reaction at the supports.
500 lb B C D E
4 ft
A
600 lb ft 3 ft 3 ft 2 ft
507
6101. The frame is used to support the 50-kg cylinder. Determine the horizontal and vertical components of reaction at A and D.
A
0.8 m
0.8 m 100 mm C B 100 mm
1.2 m
D
508
6102. The frame is used to support the 50-kg cylinder. Determine the force of the pin at C on member ABC and on member CD.
A
0.8 m
0.8 m 100 mm C B 100 mm
1.2 m
D
509
6103. Determine the reactions at the fixed support E and the smooth support A. The pin, attached to member BD, passes through a smooth slot at D.
C 600 N 0.4 m B D
0.4 m A E
0.3 m
0.3 m
0.3 m
0.3 m
510
*6104. The compound arrangement of the pan scale is shown. If the mass on the pan is 4 kg, determine the horizontal and vertical components at pins A, B, and C and the distance x of the 25-g mass to keep the scale in balance.
B
100 mm
75 mm
300 mm F E
350 mm
C
x
G 50 mm A D
4 kg
511
6105. Determine the horizontal and vertical components of reaction that the pins at A, B, and C exert on the frame. The cylinder has a mass of 80 kg.
1m C B 0.5 m A 0.7 m
100 mm D
512
6106. The bucket of the backhoe and its contents have a weight of 1200 lb and a center of gravity at G. Determine the forces of the hydraulic cylinder AB and in links AC and AD in order to hold the load in the position shown. The bucket is pinned at E.
45 A 120 C G D E
B
1 ft
0.25 ft 1.5 ft
513
6107. A man having a weight of 175 lb attempts to hold himself using one of the two methods shown. Determine the total force he must exert on bar AB in each case and the normal reaction he exerts on the platform at C. Neglect the weight of the platform.
A
B A B
C (a)
C (b)
514
*6108. A man having a weight of 175 lb attempts to hold himself using one of the two methods shown. Determine the total force he must exert on bar AB in each case and the normal reaction he exerts on the platform at C.The platform has a weight of 30 lb.
A
B A B
C (a)
C (b)
515
6109. If a clamping force of 300 N is required at A, determine the amount of force F that must be applied to the handle of the toggle clamp.
70 mm 235 mm 30 mm C 30 mm A D 30 B E 30
F
275 mm
516
6110. If a force of F = 350 N is applied to the handle of the toggle clamp, determine the resulting clamping force at A.
70 mm 235 mm 30 mm C 30 mm A D 30 B E 30
F
275 mm
517
6111. Two smooth tubes A and B, each having the same weight, W, are suspended from a common point O by means of equal-length cords. A third tube, C, is placed between A and B. Determine the greatest weight of C without upsetting equilibrium.
3r
O
3r
C
r/2
A r
B r
518
*6112. The handle of the sector press is fixed to gear G, which in turn is in mesh with the sector gear C. Note that AB is pinned at its ends to gear C and the underside of the table EF, which is allowed to move vertically due to the smooth guides at E and F. If the gears only exert tangential forces between them, determine the compressive force developed on the cylinder S when a vertical force of 40 N is applied to the handle of the press.
S E A F 40 N 0.5 m G D 1.2 m 0.2 m
H 0.35 m
B 0.65 m
C
519
6113. Show that the weight W1 of the counterweight at H required for equilibrium is W1 = (b>a)W, and so it is independent of the placement of the load W on the platform.
3b D c 4 A W E C
b F G
a H
B
c
520
6114. The tractor shovel carries a 500-kg load of soil, having a center of mass at G. Compute the forces developed in the hydraulic cylinders IJ and BC due to this loading.
A 350 mm B 200 mm C I
30
30 H J E F 300 mm 300 mm G 100 mm D
200 mm 400 mm 200 mm 50 mm
521
6115. If a force of P = 100 N is applied to the handle of the toggle clamp, determine the horizontal clamping force NE that the clamp exerts on the smooth wooden block at E.
50 mm
P 60 mm B 45 75 mm E A D 30 C 160 mm
522
*6116. If the horizontal clamping force that the toggle clamp exerts on the smooth wooden block at E is NE = 200 N, determine the force P applied to the handle of the clamp.
P 60 mm 50 mm B 45 75 mm E A D 30 C 160 mm
523
6117. The engine hoist is used to support the 200-kg engine. Determine the force acting in the hydraulic cylinder AB, the horizontal and vertical components of force at the pin C, and the reactions at the fixed support D.
G
10 350 mm C A
1250 mm
850 mm
B
550 mm D
524
6118. Determine the force that the smooth roller C exerts on member AB. Also, what are the horizontal and vertical components of reaction at pin A? Neglect the weight of the frame and roller.
60 lb ft A
C
D 0.5 ft B
3 ft
4 ft
6119. Determine the horizontal and vertical components of reaction which the pins exert on member ABC.
3 ft
525
*6120. Determine the couple moment M that must be applied to member DC for equilibrium of the quick-return mechanism. Express the result in terms of the angles f and u, dimension L, and the applied vertical force P. The block at C is confined to slide within the slot of member AB.
B
4L C M A u D P
f
L
526
6121. Determine the couple moment M that must be applied to member DC for equilibrium of the quick-return mechanism. Express the result in terms of the angles f and u, dimension L, and the applied force P, which should be changed in the figure and instead directed horizontally to the right. The block at C is confined to slide within the slot of member AB.
u
B
4L C M A D P
f
L
527
6122. The kinetic sculpture requires that each of the three pinned beams be in perfect balance at all times during its slow motion. If each member has a uniform weight of 2 lb>ft and length of 3 ft, determine the necessary counterweights W1, W2, and W3 which must be added to the ends of each member to keep the system in balance for any position. Neglect the size of the counterweights.
W
528
6123. The four-member "A" frame is supported at A and E by smooth collars and at G by a pin. All the other joints are ball-and-sockets. If the pin at G will fail when the resultant force there is 800 N, determine the largest vertical force P that can be supported by the frame. Also, what are the x, y, z force components which member BD exerts on members EDC and ABC? The collars at A and E and the pin at G only exert force components on the frame.
x
z
300 mm 300 mm E 600 mm D A 600 mm B F C G y 600 mm
P
Pk
529
*6124. The structure is subjected to the loading shown. Member AD is supported by a cable AB and roller at C and fits through a smooth circular hole at D. Member ED is supported by a roller at D and a pole that fits in a smooth snug circular hole at E. Determine the x, y, z components of reaction at E and the tension in cable AB.
z B E 0.8 m
D C x 0.5 m 0.4 m F A 0.3 m 0.3 m y
{ 2.5k} kN
530
6125. The three-member frame is connected at its ends using ball-and-socket joints. Determine the x, y, z components of reaction at B and the tension in member ED. The force acting at D is F = 5135i + 200j - 180k6 lb.
z E 6 ft 2 ft C B y D F 6 ft 3 ft x 3 ft
1ft
A 4 ft
531
6126. The structure is subjected to the loadings shown. Member AB is supported by a ball-and-socket at A and smooth collar at B. Member CD is supported by a pin at C. Determine the x, y, z components of reaction at A and C.
z 60 60 D A 4m 800 N m B
x
250 N 45
2m 1.5 m
3m
y
C
532
6127. Determine the clamping force exerted on the smooth pipe at B if a force of 20 lb is applied to the handles of the pliers. The pliers are pinned together at A.
20 lb 10 in. 40 1.5 in. B A 0.5 in. 20 lb
533
*6128. Determine the forces which the pins at A and B exert on the two-member frame which supports the 100-kg crate.
0.8 m C
0.6 m
0.4 m 0.6 m A D B
534
6129. Determine the force in each member of the truss and state if the members are in tension or compression.
E
D
C 0.1 m
3m
B
A 3m 3m
8 kN
535
6130. The space truss is supported by a ball-and-socket joint at D and short links at C and E. Determine the force in each member and state if the members are in tension or compression. Take F1 = 5 -500k6 lb and F2 = 5400j6 lb.
z C
D
F 3 ft B E x 4 ft A y F1 3 ft F2
536
537
6131. The space truss is supported by a ball-and-socket joint at D and short links at C and E. Determine the force in each member and state if the members are in tension or compression. Take F1 = 5200i + 300j - 500k6 lb and F2 = 5400j6 lb.
z C
D
F 3 ft B E x 4 ft A y F1 3 ft F2
538
539
*6132. Determine the horizontal and vertical components of reaction that the pins A and B exert on the two-member frame. Set F = 0.
1m F C
1.5 m
1m
B 60 400 N/m A
540
6133. Determine the horizontal and vertical components of reaction that pins A and B exert on the two-member frame. Set F = 500 N.
1m F C
1.5 m
1m
B 60 400 N/m A
541
6134. The two-bar mechanism consists of a lever arm AB and smooth link CD, which has a fixed smooth collar at its end C and a roller at the other end D. Determine the force P needed to hold the lever in the position u. The spring has a stiffness k and unstretched length 2L. The roller contacts either the top or bottom portion of the horizontal guide.
P B
2L C L u A k D
542
6135. Determine the horizontal and vertical components of reaction at the pin supports A and E of the compound beam assembly.
C A 1 ft B 3 ft 2 ft 1 ft D 6 ft 2 ft
2 kip/ft
E
543
*6136. Determine the force in members AB, AD, and AC of the space truss and state if the members are in tension or compression.
1.5 ft
z 1.5 ft
D 2 ft C A x B 8 ft y
F
{ 600k} lb
544
71. Determine the internal normal force and shear force, and the bending moment in the beam at points C and D. Assume the support at B is a roller. Point C is located just to the right of the 8-kip load.
A 8 ft
8 kip 40 kip ft
C 8 ft
D 8 ft
B
545
72. Determine the shear force and moment at points C and D.
A
500 lb
200 lb B C D 4 ft 6 ft
300 lb
E
6 ft
4 ft
2 ft
546
73. Determine the internal normal force, shear force, and moment at point C in the simply supported beam. Point C is located just to the right of the 1500-lb ft couple moment.
500 lb/ft
A C 6 ft 1500 lb ft 6 ft
B
30
547
*74. Determine the internal normal force, shear force, and moment at points E and F in the beam.
C
A
E
D
45
F
B
300 N/m 1.5 m 1.5 m 1.5 m 1.5 m
548
75. Determine the internal normal force, shear force, and moment at point C.
0.2 m 1m A C 1.5 m 3m 2m
400 N
B
549
76. Determine the internal normal force, shear force, and moment at point C in the simply supported beam.
4 kN/m
A C 3m 3m
B
550
77. Determine the internal normal force, shear force, and moment at point C in the cantilever beam.
w0
A L 2
C L 2
B
551
*78. Determine the internal normal force, shear force, and moment at points C and D in the simply supported beam. Point D is located just to the left of the 5-kN force.
5 kN 3 kN/m
A C 1.5 m 1.5 m D 3m
B
552
79. The bolt shank is subjected to a tension of 80 lb. Determine the internal normal force, shear force, and moment at point C.
90
C 6 in.
A
B
553
710. Determine the internal normal force, shear force, and moment at point C in the double-overhang beam.
3 kN/m
A 1.5 m
C 1.5 m 1.5 m
B 1.5 m
554
711. Determine the internal normal force, shear force, and moment at points C and D in the simply supported beam. Point D is located just to the left of the 10-kN concentrated load.
10 kN 6 kN/m
A C 1.5 m 1.5 m 1.5 m D 1.5 m
B
555
*712. Determine the internal normal force, shear force, and moment in the beam at points C and D. Point D is just to the right of the 5-kip load.
A
0.5 kip/ft
5 kip
C 6 ft 6 ft 6 ft
D 6 ft
B
556
713. Determine the internal normal force, shear force, and moment at point D of the two-member frame.
A 2m 1.5 m C E 4m D
250 N/m B
300 N/m
557
714. Determine the internal normal force, shear force, and moment at point E of the two-member frame.
A 2m 1.5 m C E 4m D
250 N/m B
300 N/m
558
715. Determine the internal normal force, shear force, and moment acting at point C and at point D, which is located just to the right of the roller support at B.
F
300 lb/ft 200 lb/ft D E A 4 ft 4 ft C 4 ft B 4 ft 200 lb/ft
*716. Determine the internal normal force, shear force, and moment in the cantilever beam at point B.
6 kip/ft
A 3 ft
B 12 ft
559
717. Determine the ratio of a>b for which the shear force will be zero at the midpoint C of the double-overhang beam.
w0
C A a b/2 C B b/2
B
a
560
718. Determine the internal normal force, shear force, and moment at points D and E in the overhang beam. Point D is located just to the left of the roller support at B, where the couple moment acts.
A
2 kN/m
6 kN m
C D 3m B 1.5 m E 1.5 m
3 5 4
5 kN
561
719. Determine the distance a in terms of the beam's length L between the symmetrically placed supports A and B so that the internal moment at the center of the beam is zero.
w0
w0
A a 2 L a 2
B
562
*720. Determine the internal normal force, shear force, and moment at points D and E in the compound beam. Point E is located just to the left of the 10-kN concentrated load. Assume the support at A is fixed and the connection at B is a pin.
2 kN/m B
10 kN
C A 1.5 m D 1.5 m 1.5 m E 1.5 m
563
721. Determine the internal normal force, shear force, and moment at points F and G in the compound beam. Point F is located just to the right of the 500-lb force, while point G is located just to the right of the 600-lb force.
500 lb 2 ft A F B C 2 ft D 1.5 ft E G 2 ft 2 ft 2 ft 600 lb
564
722. The stacker crane supports a 1.5-Mg boat with the center of mass at G. Determine the internal normal force, shear force, and moment at point D in the girder. The trolley is free to roll along the girder rail and is located at the position shown. Only vertical reactions occur at A and B.
2m A
1m1m
5m B
C
D 2m 3.5 m
7.5 m
G
565
723. Determine the internal normal force, shear force, and moment at points D and E in the two members.
1m D
0.75 m B
0.75 m A 30 60
60 N E C
2m
566
*724. Determine the internal normal force, shear force, and moment at points F and E in the frame. The crate weighs 300 lb.
A 4 ft
1.5 ft
1.5 ft
1.5 ft
1.5 ft 0.4 ft
F
C
E
D
B
567
725. Determine the internal normal force, shear force, and moment at points D and E of the frame which supports the 200-lb crate. Neglect the size of the smooth peg at C.
4.5 ft C
4 ft
E 2 ft B 1.5 ft D 1.5 ft A
568
726. The beam has a weight w per unit length. Determine the internal normal force, shear force, and moment at point C due to its weight.
B L 2
L 2
C
u A
569
727. Determine the internal normal force, shear force, and moment acting at point C. The cooling unit has a total mass of 225 kg with a center of mass at G.
F
D 0.2 m A
30
30 E C B 3m
3m
G
570
*728. The jack AB is used to straighten the bent beam DE using the arrangement shown. If the axial compressive force in the jack is 5000 lb, determine the internal moment developed at point C of the top beam. Neglect the weight of the beams.
2 ft 10 ft 10 ft 2 ft
C B A D E
571
729. Solve Prob. 728 assuming that each beam has a uniform weight of 150 lb>ft.
2 ft 10 ft 10 ft 2 ft
C B A D E
572
730. The jib crane supports a load of 750 lb from the trolley which rides on the top of the jib. Determine the internal normal force, shear force, and moment in the jib at point C when the trolley is at the position shown. The crane members are pinned together at B, E and F and supported by a short link BH.
1 ft B C
3 ft
5 ft 1 ft
3 ft G
1 ft 2 ft
H
F D E
3 ft A
750 lb
573
731. The jib crane supports a load of 750 lb from the trolley which rides on the top of the jib. Determine the internal normal force, shear force, and moment in the column at point D when the trolley is at the position shown. The crane members are pinned together at B, E and F and supported by a short link BH.
1 ft B C
3 ft
5 ft 1 ft
3 ft G
1 ft 2 ft
H
F D E
3 ft A
750 lb
574
*732. Determine the internal normal force, shear force, and moment acting at points B and C on the curved rod.
B 2 ft 30
3 4 5
C
45
A
500 lb
575
733. Determine the internal normal force, shear force, and moment at point D which is located just to the right of the 50-N force.
50 N 50 N
50 N D B 30 30
50 N
30 A 30
30 600 mm C
576
734. Determine the x, y, z components of internal loading at point C in the pipe assembly. Neglect the weight of the pipe. The load is F1 = 5 -24i - 10k6 lb, F2 = 5- 80i6 lb, and M = 5 -30k6 lb # ft.
z
F1 B M
3 ft
A
C F2
y
x
1.5 ft
2 ft
735. Determine the x, y, z components of internal loading at a section passing through point C in the pipe assembly. Neglect the weight of the pipe. Take F1 = 5350j - 400k6 lb and F2 = 5150i - 300k6 lb.
z
C x 1.5 ft 2 ft F1 3 ft
F2
y
577
*736. Determine the x, y, z components of internal loading at a section passing through point C in the pipe assembly. Neglect the weight of the pipe. Take F1 = 5- 80i + 200j - 300k6 lb and F2 = 5250i - 150j - 200k6 lb.
z
C x 1.5 ft 2 ft F1 3 ft
F2
y
578
737. The shaft is supported by a thrust bearing at A and a journal bearing at B. Determine the x, y, z components of internal loading at point C.
z A 0.2 m 750 N 0.5 m 600 N C x 1m 900 N 1m 750 N 1m 0.2 m B y
579
738. Determine the x, y, z components of internal loading in the rod at point D. There are journal bearings at A, B, and C. Take F = 57i - 12j - 5k6 kN.
0.75 m C
z
3 kN m F D B E 0.5 m x 0.2 m 0.2 m 0.6 m 0.5 m y A
580
739. Determine the x, y, z components of internal loading in the rod at point E. Take F = 57i - 12j - 5k6 kN.
z
0.75 m
C 3 kN m F D B E 0.5 m A
x
0.2 m
0.2 m 0.6 m
0.5 m y
581
*740. Draw the shear and moment diagrams for the beam (a) in terms of the parameters shown; (b) set P = 800 lb, a = 5 ft, L = 12 ft.
P
P
a L
a
582
741. Draw the shear and moment diagrams for the simply supported beam.
A 4m
9 kN
B 2m
583
584
742. Draw the shear and moment diagrams for the beam ABCDE. All pulleys have a radius of 1 ft. Neglect the weight of the beam and pulley arrangement. The load weighs 500 lb.
8 ft
2 ft
2 ft
3 ft B A 2 ft C D
2 ft E
3 ft
585
743. Draw the shear and moment diagrams for the cantilever beam.
2 kN/m
A 2m
6 kN m
586
*744. Draw the shear and moment diagrams for the beam (a) in terms of the parameters shown; (b) set M0 = 500 N # m, L = 8 m.
M0 A B L/2 L/2
587
745. If L = 9 m, the beam will fail when the maximum shear force is Vmax = 5 kN or the maximum bending moment is Mmax = 22 kN # m. Determine the largest couple moment M0 the beam will support.
M0 A B L/2 L/2
588
746. Draw the shear and moment diagrams for the simply supported beam.
A
w0
B L 2 L 2
589
590
747. Draw the shear and moment diagrams for the simply supported beam.
300 N/m 300 N m A 4m B
591
592
*748. Draw the shear and moment diagrams for the overhang beam.
A
8 kN/m
C B 4m 2m
593
749. beam.
Draw the shear and moment diagrams for the
A
2 kN/m 50 kN m B 5m 5m C
594
750.
Draw the shear and moment diagrams for the beam.
A 150 lb ft
250 lb/ ft
B 20 ft 150 lb ft
595
751.
Draw the shear and moment diagrams for the beam.
1.5 kN/m
A 3m
B
596
*752. Draw the shear and moment diagrams for the simply supported beam.
A
150 lb/ft 300 lb ft B 12 ft
597
598
753. Draw the shear and moment diagrams for the beam.
30 lb/ft 180 lb ft A B 9 ft 4.5 ft C
599
754. If L = 18 ft, the beam will fail when the maximum shear force is Vmax = 800 lb, or the maximum moment is Mmax = 1200 lb # ft. Determine the largest intensity w of the distributed loading it will support.
A L
w
B
600
755.
Draw the shear and moment diagrams for the beam.
4 kip/ft
A 12 ft 12 ft
601
*756. Draw the shear and moment diagrams for the cantilevered beam.
300 lb
200 lb/ft
A 6 ft
602
603
757. Draw the shear and moment diagrams for the overhang beam.
A B 3m
4 kN/m
3m
604
758. Determine the largest intensity w0 of the distributed load that the beam can support if the beam can withstand a maximum shear force of Vmax = 1200 lb and a maximum bending moment of Mmax = 600 lb # ft.
2w0 w0 A 6 ft 6 ft B
605
606
759. Determine the largest intensity w0 of the distributed load that the beam can support if the beam can withstand a maximum bending moment of Mmax = 20 kN # m and a maximum shear force of Vmax = 80 kN.
w0
A B 4.5 m 1.5 m
C
607
608
*760. Determine the placement a of the roller support B so that the maximum moment within the span AB is equivalent to the moment at the support B.
A
w0
B a L
609
761. The compound beam is fix supported at A, pin connected at B and supported by a roller at C. Draw the shear and moment diagrams for the beam.
A 3 ft B
500 lb/ft
C 6 ft
610
611
762. The frustum of the cone is cantilevered from point A. If the cone is made from a material having a specific weight of g, determine the internal shear force and moment in the cone as a function of x.
2 r0 A r0
L
x
612
763. Express the internal shear and moment components acting in the rod as a function of y, where 0 ... y ... 4 ft.
z
4 lb/ft x y 4 ft 2 ft
y
613
*764. Determine the normal force, shear force, and moment in the curved rod as a function of u.
w
r u
614
765. The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B. Draw the shear and moment diagrams for the shaft.
A 2 ft
600 lb 400 lb 300 lb
B 2 ft 2 ft 2 ft
615
766. Draw the shear and moment diagrams for the double overhang beam.
10 kN 5 kN 5 kN
A 2m 2m
B 2m 2m
616
767. Draw the shear and moment diagrams for the overhang beam.
A
18 kN 6 kN
B 2m 2m 2m
M = 10 kN m
617
*768. Draw the shear and moment diagrams for the simply supported beam.
M A 2m 2m 2 kN m
4 kN
B 2m
618
769. Draw the shear and moment diagrams for the simply supported beam.
A 2m
10 kN
10 kN 15 kN m B 2m 2m
619
770. Draw the shear and moment diagrams for the beam. The support at A offers no resistance to vertical load.
A
P
P
B
L 3
L 3
L 3
620
771. Draw the shear and moment diagrams for the lathe shaft if it is subjected to the loads shown. The bearing at A is a journal bearing, and B is a thrust bearing.
60 N 80 N 100 N 40 N 50 N 40 N 50 N
A 50 mm 50 mm 200 mm
B 50 mm 200 mm 100 mm 50 mm
621
*772. Draw the shear and moment diagrams for the beam.
3 kN/m
10 kN
A 6m
B
773. Draw the shear and moment diagrams for the shaft. The support at A is a thrust bearing and at B it is a journal bearing.
4 kN 2 kN/ m A B
0.8 m
0.2 m
622
774.
Draw the shear and moment diagrams for the beam.
8 kN
8 kN
15 kN/m
20 kN m A B 1m 0.25 m C 0.75 m D 1m 1m
623
775. The shaft is supported by a smooth thrust bearing at A and a smooth journal bearing at B. Draw the shear and moment diagrams for the shaft.
A
500 N 300 N/m
B 1.5 m 1.5 m
624
*776. Draw the shear and moment diagrams for the beam.
2 kN/m
10 kN
A
B
5m
3m
2m
777. Draw the shear and moment diagrams for the shaft. The support at A is a journal bearing and at B it is a thrust bearing.
200 lb A
100 lb/ft B 300 lb ft
1 ft
4 ft
1 ft
625
778. The beam consists of two segments pin connected at B. Draw the shear and moment diagrams for the beam.
A 8 ft
700 lb 150 lb/ft 800 lb ft C 6 ft
B 4 ft
626
779. Draw the shear and moment diagrams for the cantilever beam.
300 lb
200 lb/ft
A 6 ft
627
*780. Draw the shear and moment diagrams for the simply supported beam.
10 kN 10 kN/m
A
B 3m 3m
628
781. beam.
Draw the shear and moment diagrams for the
500 lb/ ft
2000 lb
A
B 9 ft 9 ft
782.
Draw the shear and moment diagrams for the beam.
A
w0
B L L
629
783.
Draw the shear and moment diagrams for the beam.
8 kN/m 3m
A 3m 8 kN/m
630
*784. Draw the shear and moment diagrams for the beam.
40 kN/m
20 kN
A B 8m 3m 150 kN m
631
785. The beam will fail when the maximum moment is Mmax = 30 kip # ft or the maximum shear is Vmax = 8 kip. Determine the largest intensity w of the distributed load the beam will support.
A 6 ft 6 ft
w B
632
786. Draw the shear and moment diagrams for the compound beam.
A
5 kN 3 kN/m
B 3m 3m
C 1.5 m 1.5 m
D
633
787. Draw the shear and moment diagrams for the shaft. The supports at A and B are journal bearings.
A
2 kN/m
B 600 mm 450 mm
300 mm
634
*788. Draw the shear and moment diagrams for the beam.
15 kip ft A 6 ft
5 kip/ft 15 kip ft B 10 ft 6 ft
635
789. Determine the tension in each segment of the cable and the cable's total length. Set P = 80 lb.
A 2 ft
B
5 ft
D C P 3 ft 4 ft 3 ft 50 lb
636
790. If each cable segment can support a maximum tension of 75 lb, determine the largest load P that can be applied.
A 2 ft
B
5 ft
D C P 3 ft 4 ft 3 ft 50 lb
637
791. The cable segments support the loading shown. Determine the horizontal distance xB from the force at B to point A. Set P = 40 lb.
5 ft
xB A
B
P
8 ft C 2 ft D 3 ft
60 lb
638
*792. The cable segments support the loading shown. Determine the magnitude of the horizontal force P so that xB = 6 ft.
5 ft
xB A
B
P
8 ft C 2 ft D 3 ft
60 lb
639
793. Determine the force P needed to hold the cable in the position shown, i.e., so segment BC remains horizontal. Also, compute the sag yB and the maximum tension in the cable.
A yB B C D
E 3m
6 kN 4 kN 4m 6m P 3m 2m
640
794. Cable ABCD supports the 10-kg lamp E and the 15-kg lamp F. Determine the maximum tension in the cable and the sag yB of point B.
A yB 2m
D
C B E F
1m
3m 0.5 m
641
795. The cable supports the three loads shown. Determine the sags yB and yD of points B and D. Take P1 = 400 lb, P2 = 250 lb.
4 ft A B C P2 P1 12 ft 20 ft 15 ft 12 ft P2 yB 14 ft yD D
E
642
*796. The cable supports the three loads shown. Determine the magnitude of P1 if P2 = 300 lb and yB = 8 ft. Also find the sag yD.
4 ft A B C P2 P1 12 ft 20 ft 15 ft 12 ft P2 yB 14 ft yD D
E
643
797. The cable supports the loading shown. Determine the horizontal distance xB the force at point B acts from A. Set P = 40 lb.
5 ft
xB A B
P
8 ft C 2 ft D
3 5 4
3 ft
30 lb
798. The cable supports the loading shown. Determine the magnitude of the horizontal force P so that xB = 6 ft.
A 5 ft
xB
B
P
8 ft C 2 ft D
3 5 4
3 ft
30 lb
644
799. Determine the maximum uniform distributed loading w0 N/m that the cable can support if it is capable of sustaining a maximum tension of 60 kN.
60 m 7m
w0
645
*7100. The cable supports the uniform distributed load of w0 = 600 lb>ft. Determine the tension in the cable at each support A and B.
A
B
15 ft 10 ft
w0 25 ft
646
7101. Determine the maximum uniform distributed load w0 the cable can support if the maximum tension the cable can sustain is 4000 lb.
10 ft
B A 15 ft
w0 25 ft
647
7102. The cable is subjected to the triangular loading. If the slope of the cable at point O is zero, determine the equation of the curve y = f1x2 which defines the cable shape OB, and the maximum tension developed in the cable.
y A
B 8 ft O x
500 lb/ ft 15 ft
500 lb/ft 15 ft
648
7103. If cylinders C and D each weigh 900 lb, determine the maximum sag h, and the length of the cable between the smooth pulleys at A and B. The beam has a weight per unit length of 100 lb>ft.
12 ft A h B
C
D
649
650
*7104. The bridge deck has a weight per unit length of 80 kN>m. It is supported on each side by a cable. Determine the tension in each cable at the piers A and B.
A 150 m
1000 m B 75 m
651
652
7105. If each of the two side cables that support the bridge deck can sustain a maximum tension of 50 MN, determine the allowable uniform distributed load w0 caused by the weight of the bridge deck.
A 150 m
1000 m B 75 m
653
654
7106. If the slope of the cable at support A is 10, determine the deflection curve y = f(x) of the cable and the maximum tension developed in the cable.
y 40 ft B A 10 10 ft x
500 lb/ft
655
7107. If h = 5 m, determine the maximum tension developed in the chain and its length. The chain has a mass per unit length of 8 kg>m.
50 m A h 5m B
656
657
*7108. A cable having a weight per unit length of 5 lb>ft is suspended between supports A and B. Determine the equation of the catenary curve of the cable and the cable's length.
150 ft A 30 30 B
658
659
7109. If the 45-m-long cable has a mass per unit length of 5 kg>m, determine the equation of the catenary curve of the cable and the maximum tension developed in the cable.
40 m A B
660
661
7110. Show that the deflection curve of the cable discussed in Example 713 reduces to Eq. 4 in Example 712 when the hyperbolic cosine function is expanded in terms of a series and only the first two terms are retained. (The answer indicates that the catenary may be replaced by a parabola in the analysis of problems in which the sag is small. In this case, the cable weight is assumed to be uniformly distributed along the horizontal.)
662
7111. The cable has a mass per unit length of 10 kg>m. Determine the shortest total length L of the cable that can be suspended in equilibrium.
8m A
B
663
664
665
*7112. The power transmission cable has a weight per unit length of 15 lb>ft. If the lowest point of the cable must be at least 90 ft above the ground, determine the maximum tension developed in the cable and the cable's length between A and B.
A
300 ft
B 180 ft 90 ft 120 ft
666
667
668
7113. If the horizontal towing force is T = 20 kN and the chain has a mass per unit length of 15 kg>m, determine the maximum sag h. Neglect the buoyancy effect of the water on the chain. The boats are stationary.
40 m T h T
669
7114. A 100-lb cable is attached between two points at a distance 50 ft apart having equal elevations. If the maximum tension developed in the cable is 75 lb, determine the length of the cable and the sag.
670
7115. Draw the shear and moment diagrams for beam CD.
3 ft
2 ft 10 kip
A C
B
4 kip ft D
2 ft
3 ft
2 ft
671
*7116. Determine the internal normal force, shear force, and moment at points B and C of the beam.
7.5 kN 2 kN/m 6 kN 1 kN/m
A
B
C 40 kN m
5m
5m 1m
3m
672
7117. Determine the internal normal force, shear force and moment at points D and E of the frame.
0.75 m
0.25 m 0.75 m C D
1m 0.75 m E B 400 N/m 60 A
673
7118. Determine the distance a between the supports in terms of the beam's length L so that the moment in the symmetric beam is zero at the beam's center.
w
a L
674
7119. A chain is suspended between points at the same elevation and spaced a distance of 60 ft apart. If it has a weight per unit length of 0.5 lb>ft and the sag is 3 ft, determine the maximum tension in the chain.
675
*7120. Draw the shear and moment diagrams for the beam.
A
2 kN/m 50 kN m B 5m 5m C
7121. Determine the internal shear and moment in member ABC as a function of x, where the origin for x is at A.
A 1.5 m B C
D 45
3m
1.5 m
1.5 m
6 kN
676
7122. The traveling crane consists of a 5-m-long beam having a uniform mass per unit length of 20 kg/m. The chain hoist and its supported load exert a force of 8 kN on the beam when x = 2 m. Draw the shear and moment diagrams for the beam. The guide wheels at the ends A and B exert only vertical reactions on the beam. Neglect the size of the trolley at C.
x A
2m
5m
C
B 8 kN
677
*7123. Determine the internal normal force, shear force, and the moment as a function of 0 ... u ... 180 and 0 ... y ... 2 ft for the member loaded as shown.
1 ft B y 150 lb 2 ft u C 200 lb
A
678
*7124. The yacht is anchored with a chain that has a total length of 40 m and a mass per unit length of 18 kg/m, and the tension in the chain at A is 7 kN. Determine the length of chain ld which is lying at the bottom of the sea. What is the distance d? Assume that buoyancy effects of the water on the chain are negligible. Hint: Establish the origin of the coordinate system at B as shown in order to find the chain length BA.
ld
A
60 d y
s B
x
679
7125. Determine the internal normal force, shear force, and moment at points D and E of the frame.
C
150 lb
30 A E 3 ft 8 ft
D F
1 ft B 4 ft
680
7126. The uniform beam weighs 500 lb and is held in the horizontal position by means of cable AB, which has a weight of 5 lb/ft. If the slope of the cable at A is 30, determine the length of the cable.
B
30 A C 15 ft
681
7127. The balloon is held in place using a 400-ft cord that weighs 0.8 lb/ft and makes a 60 angle with the horizontal. If the tension in the cord at point A is 150 lb, determine the length of the cord, l, that is lying on the ground and the height h. Hint: Establish the coordinate system at B as shown.
y
60
A
h
l
s B x
682
81. Determine the minimum horizontal force P required to hold the crate from sliding down the plane. The crate has a mass of 50 kg and the coefficient of static friction between the crate and the plane is ms = 0.25.
P
30
683
82. Determine the minimum force P required to push the crate up the plane. The crate has a mass of 50 kg and the coefficient of static friction between the crate and the plane is ms = 0.25.
P
30
684
83. A horizontal force of P = 100 N is just sufficient to hold the crate from sliding down the plane, and a horizontal force of P = 350 N is required to just push the crate up the plane. Determine the coefficient of static friction between the plane and the crate, and find the mass of the crate.
P
30
685
*84. If the coefficient of static friction at A is ms = 0.4 and the collar at B is smooth so it only exerts a horizontal force on the pipe, determine the minimum distance x so that the bracket can support the cylinder of any mass without slipping. Neglect the mass of the bracket.
100 mm x
B C 200 mm A
686
85. The 180-lb man climbs up the ladder and stops at the position shown after he senses that the ladder is on the verge of slipping. Determine the inclination u of the ladder if the coefficient of static friction between the friction pad A and the ground is ms = 0.4.Assume the wall at B is smooth.The center of gravity for the man is at G. Neglect the weight of the ladder.
B
G 10 ft
u 3 ft A
687
86. The 180-lb man climbs up the ladder and stops at the position shown after he senses that the ladder is on the verge of slipping. Determine the coefficient of static friction between the friction pad at A and ground if the inclination of the ladder is u = 60 and the wall at B is smooth.The center of gravity for the man is at G. Neglect the weight of the ladder.
B
G 10 ft
u 3 ft A
688
87. The uniform thin pole has a weight of 30 lb and a length of 26 ft. If it is placed against the smooth wall and on the rough floor in the position d = 10 ft, will it remain in this position when it is released? The coefficient of static friction is ms = 0.3.
B
26 ft
A
d
689
*88. The uniform pole has a weight of 30 lb and a length of 26 ft. Determine the maximum distance d it can be placed from the smooth wall and not slip. The coefficient of static friction between the floor and the pole is ms = 0.3.
B
26 ft
A
d
690
89. If the coefficient of static friction at all contacting surfaces is ms, determine the inclination u at which the identical blocks, each of weight W, begin to slide.
A B u
691
810. The uniform 20-lb ladder rests on the rough floor for which the coefficient of static friction is ms = 0.8 and against the smooth wall at B. Determine the horizontal force P the man must exert on the ladder in order to cause it to move.
B
5 ft
8 ft 5 ft
P
A
6 ft
692
811. The uniform 20-lb ladder rests on the rough floor for which the coefficient of static friction is ms = 0.4 and against the smooth wall at B. Determine the horizontal force P the man must exert on the ladder in order to cause it to move.
B
5 ft
8 ft 5 ft
P
A
6 ft
693
*812. The coefficients of static and kinetic friction between the drum and brake bar are ms = 0.4 and mk = 0.3, respectively. If M = 50 N # m and P = 85 N determine the horizontal and vertical components of reaction at the pin O. Neglect the weight and thickness of the brake. The drum has a mass of 25 kg.
P
300 mm B O M
700 mm
125 mm 500 mm
A
694
813. The coefficient of static friction between the drum and brake bar is ms = 0.4. If the moment M = 35 N # m, determine the smallest force P that needs to be applied to the brake bar in order to prevent the drum from rotating. Also determine the corresponding horizontal and vertical components of reaction at pin O. Neglect the weight and thickness of the brake bar. The drum has a mass of 25 kg.
P
300 mm B O M
700 mm
125 mm 500 mm
A
695
814. Determine the minimum coefficient of static friction between the uniform 50-kg spool and the wall so that the spool does not slip.
B
60
A 0.6 m
0.3 m
696
815. The spool has a mass of 200 kg and rests against the wall and on the floor. If the coefficient of static friction at B is (ms)B = 0.3, the coefficient of kinetic friction is (mk)B = 0.2, and the wall is smooth, determine the friction force developed at B when the vertical force applied to the cable is P = 800 N.
P
0.4 m G 0.1 m A
B
697
*816. The 80-lb boy stands on the beam and pulls on the cord with a force large enough to just cause him to slip. If the coefficient of static friction between his shoes and the beam is (ms)D = 0.4, determine the reactions at A and B. The beam is uniform and has a weight of 100 lb. Neglect the size of the pulleys and the thickness of the beam.
60
13 12
5
D C
A
B
5 ft 1 ft
3 ft
4 ft
698
817. The 80-lb boy stands on the beam and pulls with a force of 40 lb. If (ms)D = 0.4, determine the frictional force between his shoes and the beam and the reactions at A and B. The beam is uniform and has a weight of 100 lb. Neglect the size of the pulleys and the thickness of the beam.
60
13 12
5
D C
A
B
5 ft 1 ft
3 ft
4 ft
699
818. The tongs are used to lift the 150-kg crate, whose center of mass is at G. Determine the least coefficient of static friction at the pivot blocks so that the crate can be lifted.
500 mm
P 275 mm E C 30 F H D
500 mm A 300 mm G B
700
819. Two blocks A and B have a weight of 10 lb and 6 lb, respectively. They are resting on the incline for which the coefficients of static friction are mA = 0.15 and mB = 0.25. Determine the incline angle u for which both blocks begin to slide. Also find the required stretch or compression in the connecting spring for this to occur. The spring has a stiffness of k = 2 lb>ft.
k A u
2 lb/ft
B
701
*820. Two blocks A and B have a weight of 10 lb and 6 lb, respectively. They are resting on the incline for which the coefficients of static friction are mA = 0.15 and mB = 0.25. Determine the angle u which will cause motion of one of the blocks. What is the friction force under each of the blocks when this occurs? The spring has a stiffness of k = 2 lb>ft and is originally unstretched.
k A u
2 lb/ft
B
702
821. Crates A and B weigh 200 lb and 150 lb, respectively. They are connected together with a cable and placed on the inclined plane. If the angle u is gradually increased, determine u when the crates begin to slide. The coefficients of static friction between the crates and the plane are mA = 0.25 and mB = 0.35.
B
D A C
u
703
822. A man attempts to support a stack of books horizontally by applying a compressive force of F = 120 N to the ends of the stack with his hands. If each book has a mass of 0.95 kg, determine the greatest number of books that can be supported in the stack. The coefficient of static friction between the man's hands and a book is (ms)h = 0.6 and between any two books (ms)b = 0.4.
F
120 N
F
120 N
704
823. The paper towel dispenser carries two rolls of paper. The one in use is called the stub roll A and the other is the fresh roll B. They weigh 2 lb and 5 lb, respectively. If the coefficients of static friction at the points of contact C and D are (ms)C = 0.2 and (ms)D = 0.5, determine the initial vertical force P that must be applied to the paper on the stub roll in order to pull down a sheet.The stub roll is pinned in the center, whereas the fresh roll is not. Neglect friction at the pin.
C 4 in. 45
B
D
3 in. 60
A
P
705
*824. The drum has a weight of 100 lb and rests on the floor for which the coefficient of static friction is ms = 0.6. If a = 2 ft and b = 3 ft, determine the smallest magnitude of the force P that will cause impending motion of the drum.
P
3 5 4
a
b
825. The drum has a weight of 100 lb and rests on the floor for which the coefficient of static friction is ms = 0.5. If a = 3 ft and b = 4 ft, determine the smallest magnitude of the force P that will cause impending motion of the drum.
P
3 5 4
a
b
706
826. The refrigerator has a weight of 180 lb and rests on a tile floor for which ms = 0.25. If the man pushes horizontally on the refrigerator in the direction shown, determine the smallest magnitude of horizontal force needed to move it. Also, if the man has a weight of 150 lb, determine the smallest coefficient of friction between his shoes and the floor so that he does not slip.
4 ft
3 ft
G 1.5 ft 3 ft
A
707
827. The refrigerator has a weight of 180 lb and rests on a tile floor for which ms = 0.25. Also, the man has a weight of 150 lb and the coefficient of static friction between the floor and his shoes is ms = 0.6. If he pushes horizontally on the refrigerator, determine if he can move it. If so, does the refrigerator slip or tip?
3 ft
G 1.5 ft 4 ft 3 ft
A
708
*828. Determine the minimum force P needed to push the two 75-kg cylinders up the incline. The force acts parallel to the plane and the coefficients of static friction of the contacting surfaces are mA = 0.3, mB = 0.25, and mC = 0.4. Each cylinder has a radius of 150 mm.
P
A C B 30
709
829. If the center of gravity of the stacked tables is at G, and the stack weighs 100 lb, determine the smallest force P the boy must push on the stack in order to cause movement. The coefficient of static friction at A and B is ms = 0.3. The tables are locked together.
30 3.5 ft P A 2 ft
G 3 ft B 2 ft
710
830. The tractor has a weight of 8000 lb with center of gravity at G. Determine if it can push the 550-lb log up the incline. The coefficient of static friction between the log and the ground is ms = 0.5, and between the rear wheels of the oe tractor and the ground ms = 0.8. The front wheels are free to roll. Assume the engine can develop enough torque to cause the rear wheels to slip.
10 A
G B 2.5 ft 7 ft 3 ft 1.25 ft
C
711
831. The tractor has a weight of 8000 lb with center of gravity at G. Determine the greatest weight of the log that can be pushed up the incline. The coefficient of static friction between the log and the ground is ms = 0.5, and between the rear wheels of the tractor and the ground oe ms = 0.7. The front wheels are free to roll. Assume the engine can develop enough torque to cause the rear wheels to slip.
10
G B A 2.5 ft 7 ft 3 ft 1.25 ft
C
712
*832. The 50-kg uniform pole is on the verge of slipping at A when u = 45. Determine the coefficient of static friction at A.
C
8m
B
u 5m A
713
833. A force of P = 20 lb is applied perpendicular to the handle of the gooseneck wrecking bar as shown. If the coefficient of static friction between the bar and the wood is ms = 0.5, determine the normal force of the tines at A on the upper board. Assume the surface at C is smooth.
P 30
20 in.
1 in. A C 3 in. 3 in.
714
834. The thin rod has a weight W and rests against the floor and wall for which the coefficients of static friction are mA and mB, respectively. Determine the smallest value of u for which the rod will not move.
L
B
u A
835. A roll of paper has a uniform weight of 0.75 lb and is suspended from the wire hanger so that it rests against the wall. If the hanger has a negligible weight and the bearing at O can be considered frictionless, determine the force P needed to start turning the roll if u = 30. The coefficient of static friction between the wall and the paper is ms = 0.25.
A
30
3 in. O
u P
715
*836. A roll of paper has a uniform weight of 0.75 lb and is suspended from the wire hanger so that it rests against the wall. If the hanger has a negligible weight and the bearing at O can be considered frictionless, determine the minimum force P and the associated angle u needed to start turning the roll. The coefficient of static friction between the wall and the paper is ms = 0.25.
A
30
3 in. O
u P
716
837. If the coefficient of static friction between the chain and the inclined plane is ms = tan u, determine the overhang length b so that the chain is on the verge of slipping up the plane. The chain weighs w per unit length.
a b
u
717
838. Determine the maximum height h in meters to which the girl can walk up the slide without supporting herself by the rails or by her left leg. The coefficient of static friction between the girl's shoes and the slide is ms = 0.8.
y
y
1 x2 3
h x
718
839. If the coefficient of static friction at B is ms = 0.3, determine the largest angle u and the minimum coefficient of static friction at A so that the roller remains self-locking, regardless of the magnitude of force P applied to the belt. Neglect the weight of the roller and neglect friction between the belt and the vertical surface.
u A
30 mm
B
P
719
*840. If u = 30, determine the minimum coefficient of static friction at A and B so that the roller remains selflocking, regardless of the magnitude of force P applied to the belt. Neglect the weight of the roller and neglect friction between the belt and the vertical surface.
u A
30 mm
B
P
720
841. The clamp is used to tighten the connection between two concrete drain pipes. Determine the least coefficient of static friction at A or B so that the clamp does not slip regardless of the force in the shaft CD.
100 mm A 250 mm C B D
721
842. The coefficient of static friction between the 150-kg crate and the ground is ms = 0.3, while the coefficient of static friction between the 80-kg man's shoes and the oe ground is ms = 0.4. Determine if the man can move the crate.
30
722
843. If the coefficient of static friction between the crate and the ground is ms = 0.3, determine the minimum coefficient of static friction between the man's shoes and the ground so that the man can move the crate.
30
723
*844. The 3-Mg rear-wheel-drive skid loader has a center of mass at G. Determine the largest number of crates that can be pushed by the loader if each crate has a mass of 500 kg. The coefficient of static friction between a crate and the ground is ms = 0.3, and the coefficient of static friction between the rear wheels of the loader and the ground is oe ms = 0.5. The front wheels are free to roll. Assume that the engine of the loader is powerful enough to generate a torque that will cause the rear wheels to slip.
G 0.3 m A 0.75 m 0.25 m B
724
845. The 45-kg disk rests on the surface for which the coefficient of static friction is mA = 0.2. Determine the largest couple moment M that can be applied to the bar without causing motion.
300 mm C
M 400 mm
B 125 mm A
846. The 45-kg disk rests on the surface for which the coefficient of static friction is mA = 0.15. If M = 50 N # m, determine the friction force at A.
300 mm C
M 400 mm
B 125 mm A
725
847. Block C has a mass of 50 kg and is confined between two walls by smooth rollers. If the block rests on top of the 40-kg spool, determine the minimum cable force P needed to move the spool. The cable is wrapped around the spool's inner core. The coefficients of static friction at A and B are mA = 0.3 and mB = 0.6.
C
A 0.4 m O 0.2 m B
P
*848. Block C has a mass of 50 kg and is confined between two walls by smooth rollers. If the block rests on top of the 40-kg spool, determine the required coefficients of static friction at A and B so that the spool slips at A and B when the magnitude of the applied force is increased to P = 300 N.
C
A 0.4 m O 0.2 m B
P
726
849. The 3-Mg four-wheel-drive truck (SUV) has a center of mass at G. Determine the maximum mass of the log that can be towed by the truck. The coefficient of static friction between the log and the ground is ms = 0.8, and the coefficient of static friction between the wheels of the truck oe and the ground is ms = 0.4. Assume that the engine of the truck is powerful enough to generate a torque that will cause all the wheels to slip.
G B 0.5 m 1.6 m 1.2 m A
727
850. A 3-Mg front-wheel-drive truck (SUV) has a center of mass at G. Determine the maximum mass of the log that can be towed by the truck. The coefficient of static friction between the log and the ground is ms = 0.8, and the coefficient of static friction between the front wheels of the oe truck and the ground is ms = 0.4. The rear wheels are free to roll. Assume that the engine of the truck is powerful enough to generate a torque that will cause the front wheels to slip.
G B 0.5 m 1.6 m 1.2 m A
728
851. If the coefficients of static friction at contact points oe A and B are ms = 0.3 and ms = 0.4 respectively, determine the smallest force P that will cause the 150-kg spool to have impending motion.
400 mm
200 mm P
B 150 mm A
729
*852. If the coefficients of static friction at contact points oe A and B are ms = 0.4 and ms = 0.2 respectively, determine the smallest force P that will cause the 150-kg spool to have impending motion.
400 mm
200 mm P
B 150 mm A
730
853. The carpenter slowly pushes the uniform board horizontally over the top of the saw horse. The board has a uniform weight of 3 lb>ft, and the saw horse has a weight of 15 lb and a center of gravity at G. Determine if the saw horse will stay in position, slip, or tip if the board is pushed forward when d = 10 ft. The coefficients of static friction are shown in the figure.
18 ft d G m m 1 ft 0.5
3 ft m
0.3 1 ft
0.3
731
854. The carpenter slowly pushes the uniform board horizontally over the top of the saw horse. The board has a uniform weight of 3 lb>ft, and the saw horse has a weight of 15 lb and a center of gravity at G. Determine if the saw horse will stay in position, slip, or tip if the board is pushed forward when d = 14 ft. The coefficients of static friction are shown in the figure.
18 ft d G m m 1 ft 0.5
3 ft m
0.3 1 ft
0.3
732
855. If the 75-lb girl is at position d = 4 ft, determine the minimum coefficient of static friction ms at contact points A and B so that the plank does not slip. Neglect the weight of the plank.
A
d G
B
60
45
12 ft
733
*856. If the coefficient of static friction at the contact points A and B is ms = 0.4 , determine the minimum distance d where a 75-lb girl can stand on the plank without causing it to slip. Neglect the weight of the plank.
A
d G
B
60
45
12 ft
734
857. If each box weighs 150 lb, determine the least horizontal force P that the man must exert on the top box in order to cause motion. The coefficient of static friction between the boxes is ms = 0.5, and the coefficient of static oe friction between the box and the floor is ms = 0.2.
P
3 ft
4.5 ft
5 ft 4.5 ft A B
735
858. If each box weighs 150 lb, determine the least horizontal force P that the man must exert on the top box in order to cause motion. The coefficient of static friction between the boxes is ms = 0.65, and the coefficient of static oe friction between the box and the floor is ms = 0.35.
P
3 ft
4.5 ft
5 ft 4.5 ft A B
736
859. If the coefficient of static friction between the collars A and B and the rod is ms = 0.6, determine the maximum angle u for the system to remain in equilibrium, regardless of the weight of cylinder D. Links AC and BC have negligible weight and are connected together at C by a pin.
A 15 u u
B 15
C
D
737
*860. If u = 15, determine the minimum coefficient of static friction between the collars A and B and the rod required for the system to remain in equilibrium, regardless of the weight of cylinder D. Links AC and BC have negligible weight and are connected together at C by a pin.
A 15 u u
B 15
C
D
738
861. Each of the cylinders has a mass of 50 kg. If the coefficients of static friction at the points of contact are mA = 0.5, mB = 0.5, mC = 0.5, and mD = 0.6, determine the smallest couple moment M needed to rotate cylinder E.
A 300 mm B D M 300 mm C E
739
862. Blocks A, B, and C have weights of 50 lb, 25 lb, and 15 lb, respectively. Determine the smallest horizontal force P that will cause impending motion. The coefficient of static friction between A and B is ms = 0.3, between B and oe C, ms = 0.4, and between block C and the ground, oe m s = 0.35.
A P B C D
740
863. Determine the smallest force P that will cause impending motion. The crate and wheel have a mass of 50 kg and 25 kg, respectively. The coefficient of static friction between the crate and the ground is ms = 0.2, and oe between the wheel and the ground ms = 0.5.
P B 300 mm
C
A
741
*864. Determine the smallest force P that will cause impending motion. The crate and wheel have a mass of 50 kg and 25 kg, respectively. The coefficient of static friction between the crate and the ground is ms = 0.5, and oe between the wheel and the ground ms = 0.3.
P B 300 mm
C
A
742
865. Determine the smallest horizontal force P required to pull out wedge A. The crate has a weight of 300 lb and the coefficient of static friction at all contacting surfaces is ms = 0.3. Neglect the weight of the wedge.
P
A B 15 B
743
866. Determine the smallest horizontal force P required to lift the 200-kg crate. The coefficient of static friction at all contacting surfaces is ms = 0.3. Neglect the mass of the wedge.
A
P
B 15
744
867. Determine the smallest horizontal force P required to lift the 100-kg cylinder. The coefficients of static friction at the contact points A and B are (ms)A = 0.6 and (ms)B = 0.2, respectively; and the coefficient of static friction between the wedge and the ground is ms = 0.3.
P
0.5 m A 10 B C
745
*868. The wedge has a negligible weight and a coefficient of static friction ms = 0.35 with all contacting surfaces. Determine the largest angle u so that it is "self-locking." This requires no slipping for any magnitude of the force P applied to the joint.
u 2 P
u 2 P
746
869. Determine the smallest horizontal force P required to just move block A to the right if the spring force is 600 N and the coefficient of static friction at all contacting surfaces on A is ms = 0.3. The sleeve at C is smooth. Neglect the mass of A and B.
B
C
P
A
45
45
747
870. The three stone blocks have weights of W = 600 lb, W = 150 lb, and WC = 500 lb. Determine A B the smallest horizontal force P that must be applied to block C in order to move this block. The coefficient of static friction between the blocks is ms = 0.3, and between the oe floor and each block ms = 0.5.
45 A B C P
748
871. Determine the smallest horizontal force P required to move the wedge to the right. The coefficient of static friction at all contacting surfaces is ms = 0.3. Set u = 15 and F = 400 N. Neglect the weight of the wedge.
F A
450 mm
20 mm P C u 300 mm
B
749
*872. If the horizontal force P is removed, determine the largest angle u that will cause the wedge to be self-locking regardless of the magnitude of force F applied to the handle. The coefficient of static friction at all contacting surfaces is ms = 0.3.
F A
450 mm
20 mm P C u 300 mm
B
750
873. Determine the smallest vertical force P required to hold the wedge between the two identical cylinders, each having a weight of W. The coefficient of static friction at all contacting surfaces is ms = 0.1.
P
15
30
30
751
874. Determine the smallest vertical force P required to push the wedge between the two identical cylinders, each having a weight of W. The coefficient of static friction at all contacting surfaces is ms = 0.3.
P
15
30
30
752
875. If the uniform concrete block has a mass of 500 kg, determine the smallest horizontal force P needed to move the wedge to the left. The coefficient of static friction between the wedge and the concrete and the wedge and the floor is ms = 0.3. The coefficient of static friction between oe the concrete and floor is ms = 0.5.
150 mm A
3m
B 7.5
P
753
*876. The wedge blocks are used to hold the specimen in a tension testing machine. Determine the largest design angle u of the wedges so that the specimen will not slip regardless of the applied load. The coefficients of static friction are mA = 0.1 at A and mB = 0.6 at B. Neglect the weight of the blocks.
u
u
A
B
P
877. The square threaded screw of the clamp has a mean diameter of 14 mm and a lead of 6 mm. If ms = 0.2 for the threads, and the torque applied to the handle is 1.5 N # m, determine the compressive force F on the block.
1.5 N m
F
F
754
878. The device is used to pull the battery cable terminal C from the post of a battery. If the required pulling force is 85 lb, determine the torque M that must be applied to the handle on the screw to tighten it. The screw has square threads, a mean diameter of 0.2 in., a lead of 0.08 in., and the coefficient of static friction is ms = 0.5.
M
A
C B
879. The jacking mechanism consists of a link that has a square-threaded screw with a mean diameter of 0.5 in. and a lead of 0.20 in., and the coefficient of static friction is ms = 0.4. Determine the torque M that should be applied to the screw to start lifting the 6000-lb load acting at the end of member ABC.
6000 lb
C B M 7.5 in.
10 in. D 10 in.
A 20 in. 15 in.
755
*880. Determine the magnitude of the horizontal force P that must be applied to the handle of the bench vise in order to produce a clamping force of 600 N on the block. The single square-threaded screw has a mean diameter of 25 mm and a lead of 7.5 mm. The coefficient of static friction is ms = 0.25.
100 mm P
881. Determine the clamping force exerted on the block if a force of P = 30 N is applied to the lever of the bench vise. The single square-threaded screw has a mean diameter of 25 mm and a lead of 7.5 mm. The coefficient of static friction is ms = 0.25.
100 mm P
756
882. Determine the required horizontal force that must be applied perpendicular to the handle in order to develop a 900-N clamping force on the pipe. The single squarethreaded screw has a mean diameter of 25 mm and a lead of 5 mm. The coefficient of static friction is ms = 0.4. Note: The screw is a two-force member since it is contained within pinned collars at A and B.
E C 200 mm 150 mm A B 200 mm
D
757
883. If the clamping force on the pipe is 900 N, determine the horizontal force that must be applied perpendicular to the handle in order to loosen the screw. The single square-threaded screw has a mean diameter of 25 mm and a lead of 5 mm. The coefficient of static friction is ms = 0.4. Note: The screw is a two-force member since it is contained within pinned collars at A and B.
E C 200 mm 150 mm A B 200 mm
D
758
*884. The clamp provides pressure from several directions on the edges of the board. If the square-threaded screw has a lead of 3 mm, mean radius of 10 mm, and the coefficient of static friction is ms = 0.4, determine the horizontal force developed on the board at A and the vertical forces developed at B and C if a torque of M = 1.5 N # m is applied to the handle to tighten it further. The blocks at B and C are pin connected to the board.
B
45 A 45
D M
C
759
885. If the jack supports the 200-kg crate, determine the horizontal force that must be applied perpendicular to the handle at E to lower the crate. Each single square-threaded screw has a mean diameter of 25 mm and a lead of 7.5 mm. The coefficient of static friction is ms = 0.25.
C 45 A 45 D 45 45 B E 100 mm
760
886. If the jack is required to lift the 200-kg crate, determine the horizontal force that must be applied perpendicular to the handle at E. Each single squarethreaded screw has a mean diameter of 25 mm and a lead of 7.5 mm. The coefficient of static friction is ms = 0.25.
C 45 A 45 D 45 45 B E 100 mm
761
887. The machine part is held in place using the double-end clamp. The bolt at B has square threads with a mean radius of 4 mm and a lead of 2 mm, and the coefficient of static friction with the nut is ms = 0.5. If a torque of M = 0.4 N # m is applied to the nut to tighten it, determine the normal force of the clamp at the smooth contacts A and C.
A
260 mm
90 mm
B
C
*888. Blocks A and B weigh 50 lb and 30 lb, respectively. Using the coefficients of static friction indicated, determine the greatest weight of block D without causing motion.
m
0.5 20
B mBA A C mAC 0.6
D
0.4
762
889. Blocks A and B weigh 75 lb each, and D weighs 30 lb. Using the coefficients of static friction indicated, determine the frictional force between blocks A and B and between block A and the floor C.
m
0.5 20
B mBA A C mAC 0.6
D
0.4
890. A cylinder having a mass of 250 kg is to be supported by the cord which wraps over the pipe. Determine the smallest vertical force F needed to support the load if the cord passes (a) once over the pipe, b = 180, and (b) two times over the pipe, b = 540. Take ms = 0.2.
F
763
891. A cylinder having a mass of 250 kg is to be supported by the cord which wraps over the pipe. Determine the largest vertical force F that can be applied to the cord without moving the cylinder. The cord passes (a) once over the pipe, b = 180, and (b) two times over the pipe, b = 540. Take ms = 0.2.
F
764
*892. The boat has a weight of 500 lb and is held in position off the side of a ship by the spars at A and B. A man having a weight of 130 lb gets in the boat, wraps a rope around an overhead boom at C, and ties it to the end of the boat as shown. If the boat is disconnected from the spars, determine the minimum number of half turns the rope must make around the boom so that the boat can be safely lowered into the water at constant velocity. Also, what is the normal force between the boat and the man? The coefficient of kinetic friction between the rope and the boom is ms = 0.15. Hint: The problem requires that the normal force between the man's feet and the boat be as small as possible.
C
A
B
765
893. The 100-lb boy at A is suspended from the cable that passes over the quarter circular cliff rock. Determine if it is possible for the 185-lb woman to hoist him up; and if this is possible, what smallest force must she exert on the horizontal cable? The coefficient of static friction between the cable and the rock is ms = 0.2, and between the shoes of oe the woman and the ground ms = 0.8.
A
894. The 100-lb boy at A is suspended from the cable that passes over the quarter circular cliff rock. What horizontal force must the woman at A exert on the cable in order to let the boy descend at constant velocity? The coefficients of static and kinetic friction between the cable and the rock are ms = 0.4 and mk = 0.35, respectively.
A
766
895. A 10-kg cylinder D, which is attached to a small pulley B, is placed on the cord as shown. Determine the smallest angle u so that the cord does not slip over the peg at C. The cylinder at E has a mass of 10 kg, and the coefficient of static friction between the cord and the peg is ms = 0.1.
A
u
u
C
B E D
*896. A 10-kg cylinder D, which is attached to a small pulley B, is placed on the cord as shown. Determine the largest angle u so that the cord does not slip over the peg at C. The cylinder at E has a mass of 10 kg, and the coefficient of static friction between the cord and the peg is ms = 0.1.
A
u
u
C
B E D
767
897. Determine the smallest lever force P needed to prevent the wheel from rotating if it is subjected to a torque of M = 250 N # m. The coefficient of static friction between the belt and the wheel is ms = 0.3. The wheel is pin connected at its center, B.
400 mm B M A 200 mm
750 mm
P
768
898. If a force of P = 200 N is applied to the handle of the bell crank, determine the maximum torque M that can be resisted so that the flywheel is not on the verge of rotating clockwise. The coefficient of static friction between the brake band and the rim of the wheel is ms = 0.3.
400 mm A
P
900 mm
C 100 mm B O M 300 mm
769
899. Show that the frictional relationship between the belt tensions, the coefficient of friction m, and the angular contacts a and b for the V-belt is T2 = T1emb>sin(a>2).
Impending motion
a b
T2
T1
770
*8100. Determine the force developed in spring AB in order to hold the wheel from rotating when it is subjected to a couple moment of M = 200 N # m. The coefficient of static friction between the belt and the rim of the wheel is oe ms = 0.2, and between the belt and peg C, ms = 0.4. The pulley at B is free to rotate.
A
200 mm C 45 B
M
771
8101. If the tension in the spring is F = 2.5 kN, AB determine the largest couple moment that can be applied to the wheel without causing it to rotate. The coefficient of static friction between the belt and the wheel is ms = 0.2, oe and between the belt the peg ms = 0.4. The pulley B free to rotate.
A
200 mm C 45 B
M
772
8102. The simple band brake is constructed so that the ends of the friction strap are connected to the pin at A and the lever arm at B. If the wheel is subjected to a torque of M = 80 lb # ft, determine the smallest force P applied to the lever that is required to hold the wheel stationary. The coefficient of static friction between the strap and wheel is ms = 0.5.
M 20
80 lb ft O 45 1.25 ft B
A 1.5 ft
3 ft P
773
8103. A 180-lb farmer tries to restrain the cow from escaping by wrapping the rope two turns around the tree trunk as shown. If the cow exerts a force of 250 lb on the rope, determine if the farmer can successfully restrain the cow. The coefficient of static friction between the rope and the tree trunk is ms = 0.15, and between the farmer's shoes oe and the ground ms = 0.3.
774
*8104. The uniform 50-lb beam is supported by the rope which is attached to the end of the beam, wraps over the rough peg, and is then connected to the 100-lb block. If the coefficient of static friction between the beam and the block, and between the rope and the peg, is ms = 0.4, determine the maximum distance that the block can be placed from A and still remain in equilibrium. Assume the block will not tip.
d
1 ft A 10 ft
775
8105. The 80-kg man tries to lower the 150-kg crate using a rope that passes over the rough peg. Determine the least number of full turns in addition to the basic wrap (165) around the peg to do the job. The coefficients of static friction between the rope and the peg and between oe the man's shoes and the ground are ms = 0.1 and ms = 0.4, respectively.
15
776
8106. If the rope wraps three full turns plus the basic wrap (165) around the peg, determine if the 80-kg man can keep the 300-kg crate from moving. The coefficients of static friction between the rope and the peg and between oe the man's shoes and the ground are ms = 0.1 and ms = 0.4, respectively.
15
777
8107. The drive pulley B in a video tape recorder is on the verge of slipping when it is subjected to a torque of M = 0.005 N # m. If the coefficient of static friction between the tape and the drive wheel and between the tape and the fixed shafts A and C is ms = 0.1, determine the tensions T1 and T2 developed in the tape for equilibrium.
10 mm
T1
A 10 mm 10 mm
M B
5 mN m
C
T2
778
*8108. Determine the maximum number of 50-lb packages that can be placed on the belt without causing the belt to slip at the drive wheel A which is rotating with a constant angular velocity. Wheel B is free to rotate. Also, find the corresponding torsional moment M that must be supplied to wheel A. The conveyor belt is pre-tensioned with the 300-lb horizontal force. The coefficient of kinetic friction between the belt and platform P is mk = 0.2, and the coefficient of static friction between the belt and the rim of each wheel is ms = 0.35.
0.5 ft A M P
0.5 ft B P 300 lb
779
8109. Blocks A and B have a mass of 7 kg and 10 kg, respectively. Using the coefficients of static friction indicated, determine the largest vertical force P which can be applied to the cord without causing motion.
mB
0.4
300 mm
mD
D
0.1
B
400 mm P
mC
A C
0.4
mA
0.3
780
8110. Blocks A and B have a mass of 100 kg and 150 kg, respectively. If the coefficient of static friction between A and B and between B and C is ms = 0.25, and between the oe ropes and the pegs D and E ms = 0.5, determine the smallest force F needed to cause motion of block B if P = 30 N.
E D
A B C
45
F
P
781
8111. Block A has a weight of 100 lb and rests on a surface for which ms = 0.25. If the coefficient of static friction between the cord and the fixed peg at C is ms = 0.3, determine the greatest weight of the suspended cylinder B without causing motion.
2 ft
30 4 ft A
C
B
782
*8112. Block A has a mass of 50 kg and rests on surface B for which ms = 0.25. If the coefficient of static friction oe between the cord and the fixed peg at C is ms = 0.3, determine the greatest mass of the suspended cylinder D without causing motion.
C
3 5 4
0.25 m
A 0.3 m
0.4 m
B
D
783
8113. Block A has a mass of 50 kg and rests on surface B for which ms = 0.25. If the mass of the suspended cylinder D is 4 kg, determine the frictional force acting on A and check if motion occurs. The coefficient of static friction oe between the cord and the fixed peg at C is ms = 0.3.
C
3 5 4
0.25 m
A 0.3 m
0.4 m
B
D
8114. The collar bearing uniformly supports an axial force of P = 800 lb. If the coefficient of static friction is ms = 0.3, determine the torque M required to overcome friction.
3 in. 2 in. M P
784
8115. The collar bearing uniformly supports an axial force of P = 500 lb. If a torque of M = 3 lb # ft is applied to the shaft and causes it to rotate at constant velocity, determine the coefficient of kinetic friction at the surface of contact.
3 in. 2 in. M P
*8116. If the spring exerts a force of 900 lb on the block, determine the torque M required to rotate the shaft. The coefficient of static friction at all contacting surfaces is ms = 0.3.
2 in.
M
6 in.
8117. The disk clutch is used in standard transmissions of automobiles. If four springs are used to force the two plates A and B together, determine the force in each spring required to transmit a moment of M = 600 lb # ft across the plates. The coefficient of static friction between A and B is ms = 0.3.
A
B Fs M Fs Fs 5 in. 2 in. M
785
8118. If P = 900 N is applied to the handle of the bell crank, determine the maximum torque M the cone clutch can transmit. The coefficient of static friction at the contacting surface is ms = 0.3.
15 250 mm 300 mm C M
200 mm B 375 mm P A
786
8119. Because of wearing at the edges, the pivot bearing is subjected to a conical pressure distribution at its surface of contact. Determine the torque M required to overcome friction and turn the shaft, which supports an axial force P. The coefficient of static friction is ms. For the solution, it is necessary to determine the peak pressure p0 in terms of P and the bearing radius R.
P M
R
p0
787
*8120. The pivot bearing is subjected to a parabolic pressure distribution at its surface of contact. If the coefficient of static friction is ms, determine the torque M required to overcome friction and turn the shaft if it supports an axial force P.
P M
R r
p0
p
r2 p0 (1 ) R2
788
8121. The shaft is subjected to an axial force P. If the reactive pressure on the conical bearing is uniform, determine the torque M that is just sufficient to rotate the shaft. The coefficient of static friction at the contacting surface is ms.
d2
P
M
d1
u
u
789
8122. The tractor is used to push the 1500-lb pipe. To do this it must overcome the frictional forces at the ground, caused by sand. Assuming that the sand exerts a pressure on the bottom of the pipe as shown, and the coefficient of static friction between the pipe and the sand is ms = 0.3, determine the horizontal force required to push the pipe forward. Also, determine the peak pressure p0.
15 in. u p0 p p0 cos u
12 ft
790
8123. The conical bearing is subjected to a constant pressure distribution at its surface of contact. If the coefficient of static friction is ms, determine the torque M required to overcome friction if the shaft supports an axial force P.
P M
R
u
791
*8124. Assuming that the variation of pressure at the bottom of the pivot bearing is defined as p = p01R2>r2, determine the torque M needed to overcome friction if the shaft is subjected to an axial force P. The coefficient of static friction is ms. For the solution, it is necessary to determine p0 in terms of P and the bearing dimensions R1 and R2.
P M
R2 R1 r
p
p0 p0 R2 r
792
8125. The shaft of radius r fits loosely on the journal bearing. If the shaft transmits a vertical force P to the bearing and the coefficient of kinetic friction between the shaft and the bearing is mk, determine the torque M required to turn the shaft with constant velocity.
P
r
M
793
8126. The pulley is supported by a 25-mm-diameter pin. If the pulley fits loosely on the pin, determine the smallest force P required to raise the bucket. The bucket has a mass of 20 kg and the coefficient of static friction between the pulley and the pin is ms = 0.3. Neglect the mass of the pulley and assume that the cable does not slip on the pulley.
75 mm
z
60
P
794
8127. The pulley is supported by a 25-mm-diameter pin. If the pulley fits loosely on the pin, determine the largest force P that can be applied to the rope and yet lower the bucket. The bucket has a mass of 20 kg and the coefficient of static friction between the pulley and the pin is ms = 0.3. Neglect the mass of the pulley and assume that the cable does not slip on the pulley.
75 mm
z
60
P
795
*8128. The cylinders are suspended from the end of the bar which fits loosely into a 40-mm-diameter pin. If A has a mass of 10 kg, determine the required mass of B which is just sufficient to keep the bar from rotating clockwise. The coefficient of static friction between the bar and the pin is ms = 0.3. Neglect the mass of the bar.
A
800 mm
600 mm
B
8129. The cylinders are suspended from the end of the bar which fits loosely into a 40-mm-diameter pin. If A has a mass of 10 kg, determine the required mass of B which is just sufficient to keep the bar from rotating counterclockwise. The coefficient of static friction between the bar and the pin is ms = 0.3. Neglect the mass of the bar.
A
800 mm
600 mm
B
8130. The connecting rod is attached to the piston by a 0.75-in.-diameter pin at B and to the crank shaft by a 2-in.-diameter bearing A. If the piston is moving downwards, and the coefficient of static friction at the contact points is ms = 0.2, determine the radius of the friction circle at each connection.
B
A
796
8131. The connecting rod is attached to the piston by a 20-mm-diameter pin at B and to the crank shaft by a 50-mm-diameter bearing A. If the piston is moving upwards, and the coefficient of static friction at the contact points is ms = 0.3, determine the radius of the friction circle at each connection.
B
A
*8132. The 5-kg pulley has a diameter of 240 mm and the axle has a diameter of 40 mm. If the coefficient of kinetic friction between the axle and the pulley is mk = 0.15, determine the vertical force P on the rope required to lift the 80-kg block at constant velocity.
120 mm
P
797
8133. Solve Prob. 8132 if the force P is applied horizontally to the right.
120 mm
P
8134. The bell crank fits loosely into a 0.5-in-diameter pin. Determine the required force P which is just sufficient to rotate the bell crank clockwise. The coefficient of static friction between the pin and the bell crank is ms = 0.3.
50 lb
12 in. P
45
10 in.
798
8135. The bell crank fits loosely into a 0.5-in-diameter pin. If P = 41 lb, the bell crank is then on the verge of rotating counterclockwise. Determine the coefficient of static friction between the pin and the bell crank.
50 lb
12 in. P
45
10 in.
799
*8136. The wagon together with the load weighs 150 lb. If the coefficient of rolling resistance is a = 0.03 in., determine the force P required to pull the wagon with constant velocity.
3 in. 3 in.
P
45
800
8137. The lawn roller has a mass of 80 kg. If the arm BA is held at an angle of 30 from the horizontal and the coefficient of rolling resistance for the roller is 25 mm, determine the force P needed to push the roller at constant speed. Neglect friction developed at the axle, A, and assume that the resultant force P acting on the handle is applied along arm BA.
P B
250 mm A 30
801
8138. Determine the force P required to overcome rolling resistance and pull the 50-kg roller up the inclined plane with constant velocity. The coefficient of rolling resistance is a = 15 mm.
300 mm 30
P
30
802
8139. Determine the force P required to overcome rolling resistance and support the 50-kg roller if it rolls down the inclined plane with constant velocity. The coefficient of rolling resistance is a = 15 mm.
300 mm 30
P
30
803
*8140. The cylinder is subjected to a load that has a weight W. If the coefficients of rolling resistance for the cylinder's top and bottom surfaces are aA and aB, respectively, show that a horizontal force having a magnitude of P = [W(aA + aB)]>2r is required to move the load and thereby roll the cylinder forward. Neglect the weight of the cylinder.
W P A
r B
8141. The 1.2-Mg steel beam is moved over a level surface using a series of 30-mm-diameter rollers for which the coefficient of rolling resistance is 0.4 mm at the ground and 0.2 mm at the bottom surface of the beam. Determine the horizontal force P needed to push the beam forward at a constant speed. Hint: Use the result of Prob. 8140.
P
804
8142. Determine the smallest horizontal force P that must be exerted on the 200-lb block to move it forward. The rollers each weigh 50 lb, and the coefficient of rolling resistance at the top and bottom surfaces is a = 0.2 in.
P
1.25 ft
1.25 ft
8143. A single force P is applied to the handle of the drawer. If friction is neglected at the bottom and the coefficient of static friction along the sides is ms = 0.4, determine the largest spacing s between the symmetrically placed handles so that the drawer does not bind at the corners A and B when the force P is applied to one of the handles.
1.25 m A Chest B s P Drawer 0.3 m
805
*8144. The semicircular thin hoop of weight W and center of gravity at G is suspended by the small peg at A. A horizontal force P is slowly applied at B. If the hoop begins to slip at A when u = 30, determine the coefficient of static friction between the hoop and the peg.
A G u 2R p
R
P B
806
8145. The truck has a mass of 1.25 Mg and a center of mass at G. Determine the greatest load it can pull if (a) the truck has rear-wheel drive while the front wheels are free to roll, and (b) the truck has four-wheel drive. The coefficient of static friction between the wheels and the ground is ms = 0.5, oe and between the crate and the ground, it is ms = 0.4.
800 mm
G 600 mm A 1.5 m 1m B
807
8146. Solve Prob. 8145 if the truck and crate are traveling up a 10 incline.
600 mm
800 mm
G A 1.5 m 1m B
808
8147. If block A has a mass of 1.5 kg, determine the largest mass of block B without causing motion of the system. The coefficient of static friction between the blocks and inclined planes is ms = 0.2.
A B
45
60
809
*8148. The cone has a weight W and center of gravity at G. If a horizontal force P is gradually applied to the string attached to its vertex, determine the maximum coefficient of static friction for slipping to occur.
G
P 3 h 4
1 h 4
1 h 4
1 h 4
810
8149. The tractor pulls on the fixed tree stump. Determine the torque that must be applied by the engine to the rear wheels to cause them to slip. The front wheels are free to roll. The tractor weighs 3500 lb and has a center of gravity at G. The coefficient of static friction between the rear wheels and the ground is ms = 0.5.
G 2 ft O
2 ft A 5 ft 3 ft B
811
8150. The tractor pulls on the fixed tree stump. If the coefficient of static friction between the rear wheels and the ground is ms = 0.6, determine if the rear wheels slip or the front wheels lift off the ground as the engine provides torque to the rear wheels. What is the torque needed to cause this motion? The front wheels are free to roll. The tractor weighs 2500 lb and has a center of gravity at G.
G 2 ft O
2 ft A 5 ft 3 ft B
812
8151. A roofer, having a mass of 70 kg, walks slowly in an upright position down along the surface of a dome that has a radius of curvature of r = 20 m. If the coefficient of static friction between his shoes and the dome is ms = 0.7, determine the angle u at which he first begins to slip.
u
20 m 60
*8152. Column D is subjected to a vertical load of 8000 lb. It is supported on two identical wedges A and B for which the coefficient of static friction at the contacting surfaces between A and B and between B and C is ms = 0.4. Determine the force P needed to raise the column and the equilibrium force P needed to hold wedge A stationary. The contacting surface between A and D is smooth.
8000 lb
D 10 10 C A
P
B
P
813
8153. Column D is subjected to a vertical load of 8000 lb. It is supported on two identical wedges A and B for which the coefficient of static friction at the contacting surfaces between A and B and between B and C is ms = 0.4. If the forces P and P are removed, are the wedges self-locking? The contacting surface between A and D is smooth.
8000 lb
D 10 10 C A
P
B
P
814
91. Determine the mass and the location of the center of mass (x, y) of the uniform parabolic-shaped rod. The mass per unit length of the rod is 2 kg>m.
y
y2
4x
4m
x 4m
815
92. The uniform rod is bent into the shape of a parabola and has a weight per unit length of 6 lb>ft. Determine the reactions at the fixed support A.
y
y2
3x
3 ft
A x 3 ft
816
93. Determine the distance x to the center of mass of the homogeneous rod bent into the shape shown. If the rod has a mass per unit length of 0.5 kg>m, determine the reactions at the fixed support O.
y
1m
1m y2 O x x3
817
*94. Determine the mass and locate the center of mass (x, y) of the uniform rod. The mass per unit length of the rod is 3 kg>m.
y
y
4
x2
4m
x 2m
818
95. Determine the mass and the location of the center of mass x of the rod if its mass per unit length is m = m0(1 + x>L).
y
x L
819
96. Determine the location (x, y) of the centroid of the wire.
y
2 ft
4 ft
y
x2
x
97. Locate the centroid x of the circular rod. Express the answer in terms of the radius r and semiarc angle a.
y
r
a
C x
a
r
x
820
*98. Determine the area and the centroid (x, y) of the area.
y
4m
y2
4x
x 4m
821
99. Determine the area and the centroid (x, y) of the area.
y
1m
y2
x3
x 1m
822
910. Determine the area and the centroid (x, y) of the area.
y
3 ft
y
1 x3 9
x 3 ft
823
911. Determine the area and the centroid (x, y) of the area.
y
y2 2 ab
4ax
x b
824
*912. Locate the centroid x of the area.
y
y
x1/ 2
2x5/ 3 x
2 ft
825
913. Locate the centroid y of the area.
y
y
x1/ 2
2x5/ 3 x
2 ft
826
914. Determine the area and the centroid (x, y) of the area.
y
xy
c2 x
a b
827
915. Determine the area and the centroid (x, y) of the area.
y
h
y
h x2 a2
x a
828
*916. Locate the centroid (x, y) of the area.
y 1 1 x2 4
y 1m
x 2m
829
917. Determine the area and the centroid (x, y) of the area.
y
h y h x2 a2 x a
830
918. The plate is made of steel having a density of 7850 kg>m3. If the thickness of the plate is 10 mm, determine the horizontal and vertical components of reaction at the pin A and the tension in cable BC.
y C
B y3 2m A x 2x
4m
831
919. Determine the location x to the centroid C of the upper portion of the cardioid, r = a(1 - cos u).
r
a (1
cos u) y
C
r u _ x x
832
*920. The plate has a thickness of 0.5 in. and is made of steel having a specific weight of 490 lb>ft3. Determine the horizontal and vertical components of reaction at the pin A and the force in the cord at B.
y
B
3 ft x2 3 x 3 ft
y A
833
921.
Locate the centroid x of the shaded area.
y x2 --) 2a
y
2k(x
ka
x a
922. Locate the centroid x of the area.
y 0.5 in.
2 in.
y
1 x
0.5 in. x 2 in.
834
923. Locate the centroid y of the area.
y 0.5 in.
2 in.
y
1 x
0.5 in. x 2 in.
*924. Locate the centroid (x, y) of the area.
y
y 9 ft
9
x2
x 3 ft
835
925. Determine the area and the centroid (x, y) of the area.
y
y 3 ft
x
y
x3 9 x
3 ft
836
926. Locate the centroid x of the area.
y
y2 1m
x x2 x
y
1m
927. Locate the centroid y of the area.
y
y2 1m
x x2 x
y
1m
837
*928. Locate the centroid x of the area.
y
y
h xn an h
x a
929. Locate the centroid y of the area.
y
y
h xn an h
x a
838
930. The steel plate is 0.3 m thick and has a density of 7850 kg>m3. Determine the location of its center of mass. Also determine the horizontal and vertical reactions at the pin and the reaction at the roller support. Hint: The normal force at B is perpendicular to the tangent at B, which is found from tan u = dy>dx.
y y2 2x
2m
A
x
2m
B 2m
839
931. Locate the centroid of the area. Hint: Choose elements of thickness dy and length [(2 - y) - y2].
y
y2 1m
x
y
x
2
x 1m 1m
840
*932. Locate the centroid x of the area.
y
y2
4x
2 ft y 2x
x 1 ft
933. Locate the centroid y of the area.
y
y2
4x
2 ft y 2x
x 1 ft
841
934. If the density at any point in the rectangular plate is defined by r = r0(1 + x>a), where r0 is a constant, determine the mass and locate the center of mass x of the plate. The plate has a thickness t.
y
b 2 x b 2
a
842
935. Locate the centroid y of the homogeneous solid formed by revolving the shaded area about the y axis.
z y2 (z a)2 a2
a
y
x
843
*936. Locate the centroid z of the solid.
z
a
z
1 a (a
y)2 y
a
x
844
937. Locate the centroid y of the homogeneous solid formed by revolving the shaded area about the y axis.
z z2 1 y3 16 2m y x 4m
845
938. Locate the centroid z of the homogeneous solid frustum of the paraboloid formed by revolving the shaded area about the z axis.
z h 2 (a a2 y2)
z
h 2
h 2 y a
x
846
939. Locate the centroid y of the homogeneous solid formed by revolving the shaded area about the y axis.
z 5 ft z2 y2 9 4 ft 3 ft y
x
847
*940. Locate the center of mass y of the circular cone formed by revolving the shaded area about the y axis. The density at any point in the cone is defined by r = (r0 >h)y, where r0 is a constant.
a
z h z a y h a
y x
848
941. Determine the mass and locate the center of mass y of the hemisphere formed by revolving the shaded area about the y axis. The density at any point in the hemisphere can be defined by r = r0(1 + y>a), where r0 is a constant.
z
y2
z2
a2
r y x
849
942. Determine the volume and locate the centroid (y, z) of the homogeneous conical wedge.
z
h a y h
z x
a
y
850
943. The hemisphere of radius r is made from a stack of very thin plates such that the density varies with height, r = kz, where k is a constant. Determine its mass and the distance z to the center of mass G.
G
z
r x
_ z y
851
*944. Locate the centroid (x, y) of the uniform wire bent in the shape shown.
y 100 mm 20 mm
150 mm
x 50 mm
852
945. Locate the centroid (x, y, z) of the wire.
z
400 mm 200 mm
x
y
853
946. Locate the centroid (x, y, z) of the wire.
z
6 in.
4 in.
x
y
854
947. Locate the centroid (x, y, z) of the wire which is bent in the shape shown.
2 in.
z
2 in. 4 in.
x
y
855
*948. The truss is made from seven members, each having a mass per unit length of 6 kg/m. Locate the position (x, y) of the center of mass. Neglect the mass of the gusset plates at the joints.
y
E
D
3m
A 3m
B 3m
C
x
856
949. Locate the centroid (x, y) of the wire. If the wire is suspended from A, determine the angle segment AB makes with the vertical when the wire is in equilibrium.
y
A
60 B 200 mm 200 mm C x
857
950. Each of the three members of the frame has a mass per unit length of 6 kg/m. Locate the position (x, y) of the center of mass. Neglect the size of the pins at the joints and the thickness of the members. Also, calculate the reactions at the pin A and roller E.
6m
y 4m 4m E
C
D
B 7m
A
x
858
951. Locate the centroid (x, y) of the cross-sectional area of the channel.
y
1 in.
22 in.
x 1 in. 9 in. 1 in.
859
*952. Locate the centroid y of the cross-sectional area of the concrete beam.
12 in. 3 in.
y 12 in.
27 in.
6 in.
x
3 in. 3 in.
860
953. Locate the centroid y of the cross-sectional area of the built-up beam.
y
6 in.
1 in. 1 in.
6 in.
x 3 in. 1 in. 3 in. 1 in.
861
954. Locate the centroid y of the channel's crosssectional area.
2 in. 4 in.
2 in.
12 in.
2 in.
y C
955. Locate the distance y to the centroid of the member's cross-sectional area.
y 0.5 in. 0.5 in.
6 in.
1.5 in. 1 in. 3 in. 3 in. x
862
*956. Locate the centroid y of the cross-sectional area of the built-up beam.
y
1.5 in. 1.5 in.
4in.
4 in. 1.5 in.
3.5 in.
11.5 in. 1.5 in. x
863
957. The gravity wall is made of concrete. Determine the location (x, y) of the center of mass G for the wall.
y 1.2 m
_ x G _ y 0.4 m x 3m
2.4 m 0.6 m 0.6 m
864
958. Locate the centroid x of the composite area.
y
r0
x ri
865
959. Locate the centroid (x, y) of the composite area.
y 3 in. 4 in. 3 in.
3 in. x
866
*960. Locate the centroid (x, y) of the composite area.
y 3 ft 1.5 ft 3 ft
1 ft x
867
961. Divide the plate into parts, and using the grid for measurement, determine approximately the location (x, y) of the centroid of the plate.
y 200 mm 200 mm
x
868
962. To determine the location of the center of gravity of the automobile it is first placed in a level position, with the two wheels on one side resting on the scale platform P. In this position the scale records a reading of W1. Then, one side is elevated to a convenient height c as shown. The new reading on the scale is W2. If the automobile has a total weight of W, determine the location of its center of gravity G(x, y).
y
G
c x b W2
P
869
963. Locate the centroid y of the cross-sectional area of the built-up beam.
150 mm
y
150 mm 20 mm 200 mm
450 mm
x 20 mm
870
*964. Locate the centroid y of the cross-sectional area of the built-up beam.
200 mm
y
200 mm
20 mm 50 mm 150 mm 10 mm 300 mm 20 mm 10 mm 20 mm
x
871
965. The composite plate is made from both steel (A) and brass (B) segments. Determine the mass and location 1x, y, z2 of its mass center G. Take rst = 7.85 Mg>m3 and rbr = 8.74 Mg>m3.
A G
z
225 mm
150 mm B 150 mm 30 mm x
y
872
966. The car rests on four scales and in this position the scale readings of both the front and rear tires are shown by FA and FB. When the rear wheels are elevated to a height of 3 ft above the front scales, the new readings of the front wheels are also recorded. Use this data to compute the location x and y to the center of gravity G of the car. The tires each have a diameter of 1.98 ft.
G B _ x
_ y A
9.40 ft FB 975 lb 984 lb FA 1959 lb 1129 lb 1168 lb 2297 lb
3.0 ft
B
G
A
FA
1269 lb
1307 lb
2576 lb
873
967. Uniform blocks having a length L and mass m are stacked one on top of the other, with each block overhanging the other by a distance d, as shown. If the blocks are glued together, so that they will not topple over, determine the location x of the center of mass of a pile of n blocks.
y 2d d
x L
874
*968. Uniform blocks having a length L and mass m are stacked one on top of the other, with each block overhanging the other by a distance d, as shown. Show that the maximum number of blocks which can be stacked in this manner is n 6 L>d.
y 2d d
x L
875
969. Locate the center of gravity (x, z) of the sheetmetal bracket if the material is homogeneous and has a constant thickness. If the bracket is resting on the horizontal xy plane shown, determine the maximum angle of tilt u which it can have before it falls over, i.e., begins to rotate about the y axis.
z 60 mm 60 mm 20 mm y 20 mm 20 mm
10 mm dia. holes
80 mm 20 mm 60 mm u x
876
970. Locate the center of mass for the compressor assembly. The locations of the centers of mass of the various components and their masses are indicated and tabulated in the figure. What are the vertical reactions at blocks A and B needed to support the platform?
y 2 4 4.83 m 5 3
1 3.68 m 3.15 m 1.20 m A 1.80 m 2.30 m B
3.26 m x
2.42 m
1.19 m
2.87 m
1.64 m
1 Instrument panel 2 Filter system 3 Piping assembly 4 Liquid storage 5 Structural framework
230 kg 183 kg 120 kg 85 kg 468 kg
877
971. Major floor loadings in a shop are caused by the weights of the objects shown. Each force acts through its respective center of gravity G. Locate the center of gravity (x, y) of all these components.
z y 450 lb 1500 lb G1 9 ft 6 ft 12 ft 8 ft 4 ft 5 ft 3 ft x 7 ft 600 lb G3 G2 280 lb G4
878
*972. Locate the center of mass (x, y, z) of the homogeneous block assembly.
z
250 mm 200 mm
x
100 mm
150 mm 150 mm 150 mm y
879
973. Locate the center of mass z of the assembly. The hemisphere and the cone are made from materials having densities of 8 Mg>m3 and 4 Mg>m3, respectively.
z
100 mm
300 mm
x
y
880
974. Locate the center of mass z of the assembly. The cylinder and the cone are made from materials having densities of 5 Mg>m3 and 9 Mg>m3, respectively.
z
0.4 m
0.6 m
0.2 m
0.8 m
x
y
881
975. Locate the center of gravity (x, y, z) of the homogeneous block assembly having a hemispherical hole.
z
3 in.
1 in. 2.5 in. 2.5 in. x
1 in.
2.25 in.
3 in. 2.25 in. y
882
*976. Locate the center of gravity (x, y, z) of the assembly. The triangular and the rectangular blocks are made from materials having specific weights of 0.25 lb>in3 and 0.1 lb>in3, respectively.
z
3 in.
1 in. 2.5 in. 2.5 in. x
1 in.
2.25 in.
3 in. 2.25 in. y
883
977. Determine the distance x to the centroid of the solid which consists of a cylinder with a hole of length h = 50 mm bored into its base.
40 mm
y 120 mm
x 20 mm
h
884
978. Determine the distance h to which a hole must be bored into the cylinder so that the center of mass of the assembly is located at x = 64 mm. The material has a density of 8 Mg>m3.
40 mm
y 120 mm
x 20 mm
h
885
979. The assembly is made from a steel hemisphere, and an aluminum cylinder, rst = 7.80 Mg>m3, ral = 2.70 Mg>m3. Determine the mass center of the assembly if the height of the cylinder is h = 200 mm.
z 80 mm
G _ z
h
160 mm y
x
886
*980. The assembly is made from a steel hemisphere, and an aluminum cylinder, rst = 7.80 Mg>m3, ral = 2.70 Mg>m3. Determine the height h of the cylinder so that the mass center of the assembly is located at z = 160 mm.
z 80 mm
G _ z
h
160 mm y
x
887
981. The elevated water storage tank has a conical top and hemispherical bottom and is fabricated using thin steel plate. Determine how many square feet of plate is needed to fabricate the tank.
8 ft 6 ft 10 ft 8 ft
888
982. The elevated water storage tank has a conical top and hemispherical bottom and is fabricated using thin steel plate. Determine the volume within the tank.
8 ft 6 ft 10 ft 8 ft
889
983. Determine the volume of the solid formed by revolving the shaded area about the x axis using the second theorem of PappusGuldinus.The area and centroid y of the shaded area should first be obtained by using integration.
y 4 ft y2 4x 4 ft
x
890
*984. Determine the surface area from A to B of the tank.
B
z 3m
1.5 m A 1m
985. Determine the volume within the thin-walled tank from A to B.
B
z 3m
1.5 m A 1m
891
986. Determine the surface area of the roof of the structure if it is formed by rotating the parabola about the y axis.
y y 16 (x2/16)
16 m
x
16 m
892
987. Determine the surface area of the solid formed by revolving the shaded area 360 about the z axis.
z 0.75 in. 0.5 in. 0.75 in.
2 in. 1 in.
3 in.
893
*988. Determine the volume of the solid formed by revolving the shaded area 360 about the z axis.
z 0.75 in. 0.5 in. 0.75 in.
2 in. 1 in.
3 in.
989. Determine the volume of the solid formed by revolving the shaded area 360 about the z axis.
z 75 mm 75 mm 75 mm
250 mm 75 mm
300 mm
894
990. Determine the surface area and volume of the solid formed by revolving the shaded area 360 about the z axis.
z
1 in.
2 in.
1 in.
895
991. Determine the surface area and volume of the solid formed by revolving the shaded area 360 about the z axis.
z 75 mm 50 mm
300 mm
400 mm
75 mm
50 mm
*992. The process tank is used to store liquids during manufacturing. Estimate both the volume of the tank and its surface area. The tank has a flat top and a thin wall.
3m
3m
6m
4m
896
993. The hopper is filled to its top with coal. Estimate the volume of coal if the voids (air space) are 35 percent of the volume of the hopper.
z 1.5 m
4m
1.2 m 0.2 m
994. The thin-wall tank is fabricated from a hemisphere and cylindrical shell. Determine the vertical reactions that each of the four symmetrically placed legs exerts on the floor if the tank contains water which is 12 ft deep in the tank. The specific gravity of water is 62.4 lb>ft3. Neglect the weight of the tank.
8 ft
water surface
6 ft
4 ft
8 ft
897
995. Determine the approximate amount of paint needed to cover the outside surface of the open tank. Assume that a gallon of paint covers 400 ft2.
water surface
8 ft
6 ft
4 ft
8 ft
*996. Determine the surface area of the tank, which consists of a cylinder and hemispherical cap.
4m
8m
898
997. Determine the volume of the thin-wall tank, which consists of a cylinder and hemispherical cap.
4m
8m
899
998. The water tank AB has a hemispherical top and is fabricated from thin steel plate. Determine the volume within the tank.
B 1.6 m
1.5 m
1.6 m A 0.2 m
900
999. The water tank AB has a hemispherical roof and is fabricated from thin steel plate. If a liter of paint can cover 3 m2 of the tank's surface, determine how many liters are required to coat the surface of the tank from A to B.
B 1.6 m
1.5 m
1.6 m A 0.2 m
901
*9100. Determine the surface area and volume of the wheel formed by revolving the cross-sectional area 360 about the z axis.
z
1 in.
1 in.
1.5 in.
2 in.
4 in.
902
9101. Determine the outside surface area of the storage tank.
4 ft
15 ft
30 ft
9102. Determine the volume of the thin-wall storage tank.
4 ft
15 ft
30 ft
903
9103. Determine the height h to which liquid should be poured into the conical paper cup so that it contacts half the surface area on the inside of the cup.
100 mm
150 mm h
*9104. The tank is used to store a liquid having a specific weight of 80 lb>ft3. If it is filled to the top, determine the magnitude of the force the liquid exerts on each of its two sides ABDC and BDFE.
C A 4 ft D B F 6 ft 6 ft E 12 ft
8 ft
904
9105. The concrete "gravity" dam is held in place by its own weight. If the density of concrete is rc = 2.5 Mg>m3, and water has a density of rw = 1.0 Mg>m3, determine the smallest dimension d that will prevent the dam from overturning about its end A.
6m A d
905
9106. The symmetric concrete "gravity" dam is held in place by its own weight. If the density of concrete is rc = 2.5 Mg>m3, and water has a density of rw = 1.0 Mg>m3, determine the smallest distance d at its base that will prevent the dam from overturning about its end A. The dam has a width of 8 m.
1.5 m
9m
A d
906
9107. The tank is used to store a liquid having a specific weight of 60 lb>ft3. If the tank is full, determine the magnitude of the hydrostatic force on plates CDEF and ABDC.
z
E D F C 1.5 ft 1.5 ft x A 1.5 ft 1.5 ft y 2 ft 2 ft 5 ft B
907
*9108. The circular steel plate A is used to seal the opening on the water storage tank. Determine the magnitude of the resultant hydrostatic force that acts on it. The density of water is rw = 1 Mg>m3.
2m
45
1m A 1m B 0.5 m 0.5 m
1m
908
9109. The elliptical steel plate B is used to seal the opening on the water storage tank. Determine the magnitude of the resultant hydrostatic force that acts on it. The density of water is rw = 1 Mg>m3.
2m
45
1m A 1m B 0.5 m 0.5 m
1m
909
9110. Determine the magnitude of the hydrostatic force acting on the glass window if it is circular, A. The specific weight of seawater is gw = 63.6 lb>ft3.
4 ft 0.5 ft 1 ft A 0.5 ft B 1 ft 1 ft
910
9111. Determine the magnitude and location of the resultant hydrostatic force acting on the glass window if it is elliptical, B. The specific weight of seawater is gw = 63.6 lb>ft3.
4 ft 0.5 ft 1 ft A 0.5 ft B 1 ft 1 ft
911
*9112. Determine the magnitude of the hydrostatic force acting per foot of length on the seawall. gw = 62.4 lb>ft3.
y 2 x2
y x
8 ft
2 ft
912
9113. If segment AB of gate ABC is long enough, the gate will be on the verge of opening. Determine the length L of this segment in order for this to occur. The gate is hinged at B and has a width of 1 m. The density of water is rw = 1 Mg>m3.
A
4m L B 2m C
913
9114. If L = 2 m, determine the force the gate ABC exerts on the smooth stopper at C. The gate is hinged at B, free at A, and is 1 m wide. The density of water is rw = 1 Mg>m3.
4m L B A 2m C
914
9115. Determine the mass of the counterweight A if the 1-m-wide gate is on the verge of opening when the water is at the level shown. The gate is hinged at B and held by the smooth stop at C. The density of water is rw = 1 Mg>m3.
2m 1m B 45 2m
A
C
915
*9116. If the mass of the counterweight at A is 6500 kg, determine the force the gate exerts on the smooth stop at C. The gate is hinged at B and is 1-m wide. The density of water is rw = 1 Mg>m3.
2m 1m B 45 2m
A
C
916
9117. The concrete gravity dam is designed so that it is held in position by its own weight. Determine the factor of safety against overturning about point A if x = 2 m. The factor of safety is defined as the ratio of the stabilizing moment divided by the overturning moment. The densities of concrete and water are rconc = 2.40 Mg>m3 and rw = 1 Mg>m3, respectively.Assume that the dam does not slide.
y x 3 x2 2 6m
y
A 2m x
917
918
9118. The concrete gravity dam is designed so that it is held in position by its own weight. Determine the minimum dimension x so that the factor of safety against overturning about point A of the dam is 2. The factor of safety is defined as the ratio of the stabilizing moment divided by the overturning moment. The densities of concrete and water are rconc = 2.40 Mg>m3 and rw = 1 Mg>m3, respectively. Assume that the dam does not slide.
y x
y
3 x2 2 6m
A 2m x
919
920
9119. The underwater tunnel in the aquatic center is fabricated from a transparent polycarbonate material formed in the shape of a parabola. Determine the magnitude of the hydrostatic force that acts per meter length along the surface AB of the tunnel. The density of the water is rw = 1000 kg/m3.
y
2m y 4 x
2
A
4m
2m
2m
B
x
921
*9120.
Locate the centroid x of the shaded area.
y
y
x2 4 in.
1 in. x 1 in. 1 in.
9121.
Locate the centroid y of the shaded area.
y
y
x2 4 in.
1 in. x 1 in. 1 in.
922
9122. area.
Locate the centroid y of the beam's cross-sectional
50 mm 75 mm 25 mm
y 50 mm
75 mm
100 mm
C y x 25 mm 25 mm
9123.
Locate the centroid z of the solid.
z y2 z a a 2
2a a y
x
923
*9124. The steel plate is 0.3 m thick and has a density of 7850 kg>m3. Determine the location of its center of mass. Also compute the reactions at the pin and roller support.
y y2 2x 2m
x A 2m y x B 2m
9125.
Locate the centroid (x, y) of the area.
y
3 in.
1 in. 6 in. 3 in. x
924
9126. Determine the location (x, y) of the centroid for the structural shape. Neglect the thickness of the member.
y
3 in.
x 1.5 in. 1.5 in. 1 in. 1 in. 1.5 in. 1.5 in.
9127.
Locate the centroid y of the shaded area.
y
a
a x
a
a -- 2
a -- 2
925
*9128. The load over the plate varies linearly along the sides of the plate such that p = 2 [x(4 - y)] kPa. Determine 3 the resultant force and its position (x, y) on the plate.
p 8 kPa 3m y
4m x
9129. The pressure loading on the plate is described by the function p = 5- 240>(x + 1) + 3406 Pa. Determine the magnitude of the resultant force and coordinates of the point where the line of action of the force intersects the plate.
100 Pa
y p 300 Pa
6m 5m x
926
101. Determine the moment of inertia of the area about the x axis.
y
2m
y
0.25 x3
x 2m
927
102. Determine the moment of inertia of the area about the y axis.
y
2m
y
0.25 x3
x 2m
928
103. Determine the moment of inertia of the area about the x axis.
y
1m
y2
x3
x 1m
929
*104. Determine the moment of inertia of the area about the y axis.
y
1m
y2
x3
x 1m
930
105. Determine the moment of inertia of the area about the x axis.
y
y2 2m
2x
x 2m
931
106. Determine the moment of inertia of the area about the y axis.
y
y2 2m
2x
x 2m
932
107. Determine the moment of inertia of the area about the x axis.
y
2m
y
2x4
O
x 1m
933
*108. Determine the moment of inertia of the area about the y axis.
y
2m
y
2x4
O
x 1m
934
109. Determine the polar moment of inertia of the area about the z axis passing through point O.
y
2m
y
2x4
O
x 1m
935
1010. Determine the moment of inertia of the area about the x axis.
y
y
x3
8 in.
x 2 in.
1011. Determine the moment of inertia of the area about the y axis.
y
y
x3
8 in.
x 2 in.
936
*1012. Determine the moment of inertia of the area about the x axis.
y
y 2 in.
2 2x 3
x 1 in.
1013. Determine the moment of inertia of the area about the y axis.
y
y 2 in.
2 2x 3
x 1 in.
937
1014. Determine the moment of inertia of the area about the x axis. Solve the problem in two ways, using rectangular differential elements: (a) having a thickness of dx, and (b) having a thickness of dy.
4 in.
y
y
4 4x 2
x 1 in. 1 in.
938
1015. Determine the moment of inertia of the area about the y axis. Solve the problem in two ways, using rectangular differential elements: (a) having a thickness of dx, and (b) having a thickness of dy.
4 in.
y
y
4 4x 2
x 1 in. 1 in.
939
*1016. Determine the moment of inertia of the triangular area about the x axis.
y
y h
h (b b
x)
x b
1017. Determine the moment of inertia of the triangular area about the y axis.
y
y h
h (b b
x)
x b
940
1018. Determine the moment of inertia of the area about the x axis.
y
h
y b
h -- x2 b2 x
1019. Determine the moment of inertia of the area about the y axis.
y
h
y b
h -- x2 b2 x
941
*1020. Determine the moment of inertia of the area about the x axis.
y
2 in.
y3
x x 8 in.
942
1021. Determine the moment of inertia of the area about the y axis.
y
2 in.
y3
x x 8 in.
943
1022. Determine the moment of inertia of the area about the x axis.
y
y
2 cos ( x) 8
2 in. x 4 in. 4 in.
1023. Determine the moment of inertia of the area about the y axis.
y
y
2 cos ( x) 8
2 in. x 4 in. 4 in.
944
*1024. Determine the moment of inertia of the area about the x axis.
y x2 y2
2 r0
r0 x
945
1025. Determine the moment of inertia of the area about the y axis.
y x2 y2
2 r0
r0 x
946
1026. Determine the polar moment of inertia of the area about the z axis passing through point O.
y x2 y2
2 r0
r0 x
1027. Determine the distance y to the centroid of the beam's cross-sectional area; then find the moment of inertia about the x axis.
y 6 in. x y C 2 in. x 4 in.
1 in.
1 in.
947
*1028. Determine the moment of inertia of the beam's cross-sectional area about the x axis.
y 6 in. x y C 2 in. x 4 in.
1 in.
1 in.
1029. Determine the moment of inertia of the beam's cross-sectional area about the y axis.
y 6 in. x y C 2 in. x 4 in.
1 in.
1 in.
948
1030. Determine the moment of inertia of the beam's cross-sectional area about the x axis.
15 mm
y 60 mm 60 mm 15 mm
100 mm 15 mm 50 mm x 50 mm 100 mm 15 mm
949
1031. Determine the moment of inertia of the beam's cross-sectional area about the y axis.
15 mm
y 60 mm 60 mm 15 mm
100 mm 15 mm 50 mm x 50 mm 100 mm 15 mm
950
*1032. Determine the moment of inertia of the composite area about the x axis.
y
150 mm 150 mm
100 mm 100 mm x 300 mm 75 mm
951
1033. Determine the moment of inertia of the composite area about the y axis.
y
150 mm 150 mm
100 mm 100 mm x 300 mm 75 mm
952
1034. Determine the distance y to the centroid of the beam's cross-sectional area; then determine the moment of inertia about the x axis.
y 25 mm 25 mm
C _ y 75 mm 75 mm
100 mm x 25 mm x 50 mm
50 mm 100 mm
25 mm
953
1035. Determine the moment of inertia of the beam's cross-sectional area about the y axis.
y 25 mm 25 mm
C _ y 75 mm 75 mm
100 mm x 25 mm x 50 mm
50 mm 100 mm
25 mm
*1036. Locate the centroid y of the composite area, then determine the moment of inertia of this area about the centroidal x axis.
1 in. 5 in. 2 in.
y
1 in.
C
x y x
3 in.
3 in.
954
1037. Determine the moment of inertia of the composite area about the centroidal y axis.
y
1 in. 5 in. 2 in.
1 in.
C
x y x
3 in.
3 in.
1038. Determine the distance y to the centroid of the beam's cross-sectional area; then find the moment of inertia about the x axis.
y 50 mm 50 mm
300 mm C y 100 mm x 200 mm x
955
1039. Determine the moment of inertia of the beam's cross-sectional area about the x axis.
y 50 mm 50 mm
300 mm C y 100 mm x 200 mm x
956
*1040. Determine the moment of inertia of the beam's cross-sectional area about the y axis.
y 50 mm 50 mm
300 mm C y 100 mm x 200 mm x
957
1041. Determine the moment of inertia of the beam's cross-sectional area about the x axis.
y
15 mm 115 mm 7.5 mm x 115 mm 15 mm
50 mm 50 mm
958
1042. Determine the moment of inertia of the beam's cross-sectional area about the y axis.
y
15 mm 115 mm 7.5 mm x 115 mm 15 mm
50 mm 50 mm
959
1043. Locate the centroid y of the cross-sectional area for the angle. Then find the moment of inertia Ix about the x centroidal axis.
y x
y
6 in.
C 2 in. 2 in. 6 in. y
x
x
*1044. Locate the centroid x of the cross-sectional area for the angle. Then find the moment of inertia Iy about the y centroidal axis.
y x
y
6 in.
C 2 in. 2 in. 6 in. y
x
x
960
1045. Determine the moment of inertia of the composite area about the x axis.
y
150 mm x 150 mm
150 mm
150 mm
961
1046. Determine the moment of inertia of the composite area about the y axis.
y
150 mm x 150 mm
150 mm
150 mm
962
1047. Determine the moment of inertia of the composite area about the centroidal y axis.
240 mm
y
50 mm C 400 mm 50 mm y x
x 150 mm 150 mm 50 mm
963
*1048. Locate the centroid y of the composite area, then determine the moment of inertia of this area about the x axis.
240 mm
y
50 mm C 400 mm 50 mm y x
x 150 mm 150 mm 50 mm
964
1049. Determine the moment of inertia Ix of the section. The origin of coordinates is at the centroid C.
y
200 mm x 20 mm 20 mm 20 mm C 600 mm 200 mm
1050. Determine the moment of inertia Iy of the section. The origin of coordinates is at the centroid C.
y
200 mm x 20 mm 20 mm 20 mm C 600 mm 200 mm
965
1051. Determine the beam's moment of inertia Ix about the centroidal x axis.
15 mm
y 15 mm
50 mm 50 mm C x 10 mm 100 mm
100 mm
966
*1052. Determine the beam's moment of inertia Iy about the centroidal y axis.
15 mm
y 15 mm
50 mm 50 mm C x 10 mm 100 mm
100 mm
967
1053. Locate the centroid y of the channel's crosssectional area, then determine the moment of inertia of the area about the centroidal x axis.
0.5 in.
y
x 6 in. C y x 6.5 in. 0.5 in. 6.5 in. 0.5 in.
968
1054. Determine the moment of inertia of the area of the channel about the y axis.
0.5 in.
y
x 6 in. C y x 6.5 in. 0.5 in. 6.5 in. 0.5 in.
969
1055. Determine the moment of inertia of the crosssectional area about the x axis.
10 mm
y x
y
180 mm C 100 mm 10 mm 10 mm 100 mm
x
970
*1056. Locate the centroid x of the beam's crosssectional area, and then determine the moment of inertia of the area about the centroidal y axis.
10 mm
y x
y
180 mm C 100 mm 10 mm 10 mm 100 mm
x
971
1057. Determine the moment of inertia of the beam's cross-sectional area about the x axis.
y 125 mm 125 mm 12 mm 12 mm 75 mm x 75 mm
12 mm 100 mm 25 mm 12 mm
972
1058. Determine the moment of inertia of the beam's cross-sectional area about the y axis.
125 mm
y 125 mm 12 mm 12 mm 75 mm x 75 mm
12 mm 100 mm 25 mm 12 mm
973
1059. Determine the moment of inertia of the beam's cross-sectional area with respect to the x axis passing through the centroid C of the cross section. y = 104.3 mm.
35 mm A
150 mm C x 15 mm y B 50 mm
*1060. Determine the product of inertia of the parabolic area with respect to the x and y axes.
y
1 in.
2 in. y 2x2
x
974
1061. Determine the product of inertia Ixy of the right half of the parabolic area in Prob. 1060, bounded by the lines y = 2 in. and x = 0.
y
1 in.
2 in. y 2x2
x
975
1062. Determine the product of inertia of the quarter elliptical area with respect to the x and y axes.
y
2 2
x a2 b
y b2
1
x a
976
1063. Determine the product of inertia for the area with respect to the x and y axes.
2 in.
y
y3
x x 8 in.
977
*1064. Determine the product of inertia of the area with respect to the x and y axes.
y
4 in. x
4 in.
y
x (x 4
8)
978
1065. Determine the product of inertia of the area with respect to the x and y axes.
y
8y
x3
2x2
4x
3m
x 2m
979
1066. Determine the product of inertia for the area with respect to the x and y axes.
y y2 1 0.5x
1m
x 2m
980
1067. Determine the product of inertia for the area with respect to the x and y axes.
y
y3
h3 x b h x b
981
*1068. Determine the product of inertia for the area of the ellipse with respect to the x and y axes.
y
x2
4y2
16
2 in.
x 4 in.
1069. Determine the product of inertia for the parabolic area with respect to the x and y axes.
y
y2
x 2 in. x
4 in.
982
1070. Determine the product of inertia of the composite area with respect to the x and y axes.
y
2 in.
2 in.
2 in.
1.5 in. 2 in. x
983
1071. Determine the product of inertia of the crosssectional area with respect to the x and y axes that have their origin located at the centroid C.
y 4 in.
1 in. 0.5 in. 5 in. C 3.5 in. 1 in. 4 in. x
*1072. Determine the product of inertia for the beam's cross-sectional area with respect to the x and y axes that have their origin located at the centroid C.
y 5 mm
50 mm
7.5 mm C 17.5 mm 5 mm 30 mm x
984
1073. Determine the product of inertia of the beam's cross-sectional area with respect to the x and y axes.
y 10 mm
300 mm
10 mm x 10 mm 100 mm
1074. Determine the product of inertia for the beam's cross-sectional area with respect to the x and y axes that have their origin located at the centroid C.
y
5 in.
1 in. C
0.5 in. x 5 in. 5 in.
1 in.
5 in.
1 in.
985
1075. Locate the centroid x of the beam's cross-sectional area and then determine the moments of inertia and the product of inertia of this area with respect to the u and v axes. The axes have their origin at the centroid C.
y x 20 mm v 200 mm C 60 200 mm 20 mm x
20 mm 175 mm
u
986
*1076. Locate the centroid (x, y) of the beam's crosssectional area, and then determine the product of inertia of this area with respect to the centroidal x and y axes.
y x
y
10 mm
100 mm 10 mm 300 mm C 10 mm x 200 mm x y
987
1077. Determine the product of inertia of the beam's cross-sectional area with respect to the centroidal x and y axes.
y 100 mm 5 mm 10 mm 150 mm 10 mm C 150 mm x
100 mm
10 mm
988
1078. Determine the moments of inertia and the product of inertia of the beam's cross-sectional area with respect to the u and v axes.
y v 1.5 in. u
1.5 in.
3 in. 30 C 3 in. x
989
1079. Locate the centroid y of the beam's cross-sectional area and then determine the moments of inertia and the product of inertia of this area with respect to the u and v axes.
y u v 0.5 in. 4.5 in. 0.5 in. 4 in. 0.5 in. 8 in. y C
4.5 in.
60 x
990
991
*1080. Locate the centroid x and y of the cross-sectional area and then determine the orientation of the principal axes, which have their origin at the centroid C of the area. Also, find the principal moments of inertia.
y x
0.5 in. 6 in. C 0.5 in. y x
6 in.
992
993
1081. Determine the orientation of the principal axes, which have their origin at centroid C of the beam's crosssectional area. Also, find the principal moments of inertia.
100 mm 20 mm 20 mm
y
150 mm x 150 mm
C
100 mm
20 mm
994
995
1082. Locate the centroid y of the beam's cross-sectional area and then determine the moments of inertia of this area and the product of inertia with respect to the u and v axes. The axes have their origin at the centroid C.
y
25 mm
25 mm
v
200 mm
C 60 y
x
25 mm 75 mm 75 mm u
996
997
1083.
Solve Prob. 1075 using Mohr's circle.
998
*1084.
Solve Prob. 1078 using Mohr's circle.
999
1085.
Solve Prob. 1079 using Mohr's circle.
1000
1086.
Solve Prob. 1080 using Mohr's circle.
1001
1087.
Solve Prob. 1081 using Mohr's circle.
1002
*1088.
Solve Prob. 1082 using Mohr's circle.
1003
1089. Determine the mass moment of inertia Iz of the cone formed by revolving the shaded area around the z axis. The density of the material is r. Express the result in terms of the mass m of the cone.
z z h (r0 r0 y)
h
y x r0
1004
1090. Determine the mass moment of inertia Ix of the right circular cone and express the result in terms of the total mass m of the cone. The cone has a constant density r.
y
y
r x h
r x
h
1005
1091. Determine the mass moment of inertia Iy of the slender rod. The rod is made of material having a variable density r = r0(1 + x>l), where r0 is constant. The crosssectional area of the rod is A. Express the result in terms of the mass m of the rod.
z
l y
x
1006
*1092. Determine the mass moment of inertia Iy of the solid formed by revolving the shaded area around the y axis. The density of the material is r. Express the result in terms of the mass m of the solid.
z
z
1 y2 4 1m y
x 2m
1007
1093. The paraboloid is formed by revolving the shaded area around the x axis. Determine the radius of gyration kx. The density of the material is r = 5 Mg>m3.
y y2 50 x 100 mm x
200 mm
1008
1094. Determine the mass moment of inertia Iy of the solid formed by revolving the shaded area around the y axis. The density of the material is r. Express the result in terms of the mass m of the semi-ellipsoid.
b
z a y2 a2 z2 b2 1
y
x
1009
1095. The frustum is formed by rotating the shaded area around the x axis. Determine the moment of inertia Ix and express the result in terms of the total mass m of the frustum. The material has a constant density r.
b
y y
b x a
b 2b x
a
1010
*1096. The solid is formed by revolving the shaded area around the y axis. Determine the radius of gyration ky. The specific weight of the material is g = 380 lb>ft3.
y
3 in. y3 9x x 3 in.
1011
1097. Determine the mass moment of inertia Iz of the solid formed by revolving the shaded area around the z axis. The density of the material is r = 7.85 Mg>m3.
z
2m
z2
8y
4m
y
x
1012
1098. Determine the mass moment of inertia Iz of the solid formed by revolving the shaded area around the z axis. The solid is made of a homogeneous material that weighs 400 lb.
z
4 ft
8 ft z y2
3
y
x
1013
1099. Determine the mass moment of inertia Iy of the solid formed by revolving the shaded area around the y axis. The total mass of the solid is 1500 kg.
z
4m
z2
1 y3 16
2m
O
y
x
1014
*10100. Determine the mass moment of inertia of the pendulum about an axis perpendicular to the page and passing through point O. The slender rod has a mass of 10 kg and the sphere has a mass of 15 kg.
O
450 mm
A
100 mm B
1015
10101. The pendulum consists of a disk having a mass of 6 kg and slender rods AB and DC which have a mass per unit length of 2 kg>m. Determine the length L of DC so that the center of mass is at the bearing O. What is the moment of inertia of the assembly about an axis perpendicular to the page and passing through point O?
0.8 m 0.2 m A O
0.5 m
D
L B C
1016
10102. Determine the mass moment of inertia of the 2-kg bent rod about the z axis.
z
300 mm
x
300 mm
y
1017
10103. The thin plate has a mass per unit area of 10 kg>m2. Determine its mass moment of inertia about the y axis.
z 200 mm 200 mm 100 mm
200 mm 100 mm 200 mm 200 mm x 200 mm 200 mm 200 mm y
1018
*10104. The thin plate has a mass per unit area of 10 kg>m2. Determine its mass moment of inertia about the z axis.
z 200 mm 200 mm 100 mm
200 mm 100 mm 200 mm 200 mm x 200 mm 200 mm 200 mm y
1019
10105. The pendulum consists of the 3-kg slender rod and the 5-kg thin plate. Determine the location y of the center of mass G of the pendulum; then find the mass moment of inertia of the pendulum about an axis perpendicular to the page and passing through G.
y
O
2m
G 0.5 m 1m
1020
10106. The cone and cylinder assembly is made of homogeneous material having a density of 7.85 Mg>m3. Determine its mass moment of inertia about the z axis.
z
150 mm
300 mm
150 mm 300 mm x
y
1021
10107. Determine the mass moment of inertia of the overhung crank about the x axis. The material is steel having a density of r = 7.85 Mg>m3.
20 mm 30 mm 90 mm 50 mm x 20 mm x 30 mm 20 mm 50 mm 30 mm 180 mm
1022
*10108. Determine the mass moment of inertia of the overhung crank about the x axis. The material is steel having a density of r = 7.85 Mg>m3.
20 mm 30 mm 90 mm 50 mm x 20 mm x 30 mm 20 mm 50 mm 30 mm 180 mm
1023
10109. If the large ring, small ring and each of the spokes weigh 100 lb, 15 lb, and 20 lb, respectively, determine the mass moment of inertia of the wheel about an axis perpendicular to the page and passing through point A.
1 ft
4 ft
O
A
1024
10110. Determine the mass moment of inertia of the thin plate about an axis perpendicular to the page and passing through point O. The material has a mass per unit area of 20 kg>m2.
50 mm
O
150 mm 50 mm
150 mm 400 mm
400 mm
150 mm 150 mm
1025
10111. Determine the mass moment of inertia of the thin plate about an axis perpendicular to the page and passing through point O. The material has a mass per unit area of 20 kg>m2.
O 200 mm
200 mm
200 mm
1026
*10112. Determine the moment of inertia of the beam's cross-sectional area about the x axis which passes through the centroid C.
d 2 d 2
y
60 60 d 2 C x
d 2
10113. Determine the moment of inertia of the beam's cross-sectional area about the y axis which passes through the centroid C.
d 2 d 2
y
60 60 d 2 C x
d 2
1027
10114. Determine the moment of inertia of the beam's cross-sectional area about the x axis.
y
y a a
a x 2
x
a
a
1028
10115. Determine the moment of inertia of the beam's cross-sectional area with respect to the x axis passing through the centroid C.
0.5 in.
4 in.
_ y 2.5 in. C x
0.5 in.
0.5 in.
*10116. Determine the product of inertia for the angle's cross-sectional area with respect to the x and y axes having their origin located at the centroid C. Assume all corners to be right angles.
y 57.37 mm 20 mm
200 mm C 20 mm 200 mm x 57.37 mm
1029
10117. Determine the moment of inertia of the area about the y axis.
y
4y 1 ft
4 x2
x 2 ft
10118. Determine the moment of inertia of the area about the x axis.
y
4y 1 ft
4 x2
x 2 ft
1030
10119. Determine the moment of inertia of the area about the x axis. Then, using the parallel-axis theorem, find the moment of inertia about the x axis that passes through the centroid C of the area. y = 120 mm.
y 200 mm
200 mm y
C y
x
1 x 2 200
x
1031
*10120. The pendulum consists of the slender rod OA, which has a mass per unit length of 3 kg>m. The thin disk has a mass per unit area of 12 kg>m2. Determine the distance y to the center of mass G of the pendulum; then calculate the moment of inertia of the pendulum about an axis perpendicular to the page and passing through G.
O
y 1.5 m
G A 0.1 m 0.3 m
1032
10121. Determine the product of inertia of the area with respect to the x and y axes.
y
1m y 1m x3 x
1033
111. The 200-kg crate is on the lift table at the position u = 30. Determine the force in the hydraulic cylinder AD for equilibrium. Neglect the mass of the lift table's components.
I F 1.2 m
H
1.2 m
D
E
C A
u B
1034
112. The uniform rod OA has a weight of 10 lb. When the rod is in a vertical position, u = 0, the spring is unstretched. Determine the angle u for equilibrium if the end of the spring wraps around the periphery of the disk as the disk turns.
A
2 ft
u
O 0.5 ft
k
30 lb/ft
1035
113. The "Nuremberg scissors" is subjected to a horizontal force of P = 600 N. Determine the angle u for equilibrium. The spring has a stiffness of k = 15 kN>m and is unstretched when u = 15.
E u A k 200 mm B 200 mm D C P
1036
*114. The "Nuremberg scissors" is subjected to a horizontal force of P = 600 N. Determine the stiffness k of the spring for equilibrium when u = 60. The spring is unstretched when u = 15.
E u A k 200 mm B 200 mm D C P
1037
115. Determine the force developed in the spring required to keep the 10 lb uniform rod AB in equilibrium when u = 35.
k
15 lb/ft B
M = 10 lb ft 6 ft u A
1038
116. If a force of P = 5 lb is applied to the handle of the mechanism, determine the force the screw exerts on the cork of the bottle. The screw is attached to the pin at A and passes through the collar that is attached to the bottle neck at B.
P
5 lb
D
A u 3 in. B 30
1039
117. The pin-connected mechanism is constrained at A by a pin and at B by a roller. If P = 10 lb, determine the angle u for equilibrium. The spring is unstretched when u = 45. Neglect the weight of the members.
A
B u 0.5 ft
P
k 0.5 ft
50 lb/ft 0.5 ft
1040
*118. The pin-connected mechanism is constrained by a pin at A and a roller at B. Determine the force P that must be applied to the roller to hold the mechanism in equilibrium when u = 30. The spring is unstretched when u = 45. Neglect the weight of the members.
A
B u 0.5 ft
P
k 0.5 ft
50 lb/ft 0.5 ft
1041
119. If a force P = 100 N is applied to the lever arm of the toggle press, determine the clamping force developed in the block when u = 45. Neglect the weight of the block.
F D u u A B
E 200 mm
P
C
200 mm 500 mm
1042
1110. When the forces are applied to the handles of the bottle opener, determine the pulling force developed on the cork.
P
5N
P
5N
90 mm A B 15 mm
90 mm E D 15 mm C
F
1043
1111. If the spring has a stiffness k and an unstretched length l0, determine the force P when the mechanism is in the position shown. Neglect the weight of the members.
l u
A
P
B
k
l
C
*1112. Solve Prob. 1111 if the force P is applied vertically downward at B.
l
A
u
P
B
k
l
C
1044
1113. Determine the angles u for equilibrium of the 4-lb disk using the principle of virtual work. Neglect the weight of the rod. The spring is unstretched when u = 0 and always remains in the vertical position due to the roller guide.
1 ft 3 ft B A u C
k
50 lb/ft
1045
1114. The truck is weighed on the highway inspection scale. If a known mass m is placed a distance s from the fulcrum B of the scale, determine the mass of the truck mt if its center of gravity is located at a distance d from point C. When the scale is empty, the weight of the lever ABC balances the scale CDE.
A
d
s
a C B E
m
a
D
1046
1115. The assembly is used for exercise. It consists of four pin-connected bars, each of length L, and a spring of stiffness k and unstretched length a (6 2L). If horizontal forces are applied to the handles so that u is slowly decreased, determine the angle u at which the magnitude of P becomes a maximum.
P B
A
L u k u
L
P D
L
L
C
1047
*1116. A 5-kg uniform serving table is supported on each side by pairs of two identical links, AB and CD, and springs CE. If the bowl has a mass of 1 kg, determine the angle u where the table is in equilibrium. The springs each have a stiffness of k = 200 N>m and are unstretched when u = 90. Neglect the mass of the links.
250 mm
150 mm
A 250 mm u
E C k
u D B 150 mm
1048
1117. A 5-kg uniform serving table is supported on each side by two pairs of identical links, AB and CD, and springs CE. If the bowl has a mass of 1 kg and is in equilibrium when u = 45, determine the stiffness k of each spring. The springs are unstretched when u = 90. Neglect the mass of the links.
250 mm
150 mm
A 250 mm u
E C k
u D B 150 mm
1049
1118. If a vertical force of P = 50 N is applied to the handle of the toggle clamp, determine the clamping force exerted on the pipe.
100 mm 300 mm A 150 mm u B C 500 mm
P
50 N
45
D
1050
1119. The spring is unstretched when u = 45 and has a stiffness of k = 1000 lb>ft. Determine the angle u for equilibrium if each of the cylinders weighs 50 lb. Neglect the weight of the members. The spring remains horizontal at all times due to the roller.
4 ft 2 ft u k B 4 ft E A u D
2 ft
C
1051
*1120. The machine shown is used for forming metal plates. It consists of two toggles ABC and DEF, which are operated by the hydraulic cylinder. The toggles push the moveable bar G forward, pressing the plate into the cavity. If the force which the plate exerts on the head is P = 8 kN, determine the force F in the hydraulic cylinder when u = 30.
D 200 mm
u
30 E H
F 200 mm
plate
F F 200 mm A u B
P
200 mm C G
1052
1121. The vent plate is supported at B by a pin. If it weighs 15 lb and has a center of gravity at G, determine the stiffness k of the spring so that the plate remains in equilibrium at u = 30. The spring is unstretched when u = 0.
u G B k 4 ft A 0.5 ft 1 ft
C
1053
1122. Determine the weight of block G required to balance the differential lever when the 20-lb load F is placed on the pan. The lever is in balance when the load and block are not on the lever. Take x = 12 in.
4 in. A B D 2 in.
4 in. C
x G
E
F
1054
1123. If the load F weighs 20 lb and the block G weighs 2 lb, determine its position x for equilibrium of the differential lever. The lever is in balance when the load and block are not on the lever.
4 in. A B D 2 in.
4 in. C
x G
E
F
1055
*1124. Determine the magnitude of the couple moment M required to support the 20-kg cylinder in the configuration shown. The smooth peg at B can slide freely within the slot. Neglect the mass of the members.
2.5 m B
D
1m A u 30 M C
E
1m
1056
1125. The crankshaft is subjected to a torque of M = 50 lb # ft. Determine the vertical compressive force F applied to the piston for equilibrium when u = 60.
F
5 in.
B u M
3 in.
A
1057
1126. If the potential energy for a conservative onedegree-of-freedom system is expressed by the relation V = (4x3 - x2 - 3x + 10) ft # lb, where x is given in feet, determine the equilibrium positions and investigate the stability at each position.
1127. If the potential energy for a conservative onedegree-of-freedom system is expressed by the relation V = (24 sin u + 10 cos 2u) ft # lb, 0 ... u ... 90, determine the equilibrium positions and investigate the stability at each position.
1058
*1128. If the potential energy for a conservative onedegree-of-freedom system is expressed by the relation V = (3y3 + 2y2 - 4y + 50) J, where y is given in meters, determine the equilibrium positions and investigate the stability at each position.
1059
1129. The 2-Mg bridge, with center of mass at point G, is lifted by two beams CD, located at each side of the bridge. If the 2-Mg counterweight E is attached to the beams as shown, determine the angle u for equilibrium. Neglect the weight of the beams and the tie rods.
5m C 0.3 m A 2.5 m u D 2m E G u 2.5 m B 2m
1060
1130. The spring has a stiffness k = 600 lb>ft and is unstretched when u = 45. If the mechanism is in equilibrium when u = 60, determine the weight of cylinder D. Neglect the weight of the members. Rod AB remains horizontal at all times since the collar can slide freely along the vertical guide.
k B A
u C
5 ft
D
1061
1131. If the springs at A and C have an unstretched length of 10 in. while the spring at B has an unstretched length of 12 in., determine the height h of the platform when the system is in equilibrium. Investigate the stability of this equilibrium configuration. The package and the platform have a total weight of 150 lb.
h
A
B
C
k1
20 lb/in.
k1
20 lb/in.
k2
30 lb/in.
1062
*1132. The spring is unstretched when u = 45 and has a stiffness of k = 1000 lb>ft. Determine the angle u for equilibrium if each of the cylinders weighs 50 lb. Neglect the weight of the members.
4 ft 2 ft 2 ft k B 4 ft E u A u D
C
1063
1133. A 5-kg uniform serving table is supported on each side by pairs of two identical links, AB and CD, and springs CE. If the bowl has a mass of 1 kg, determine the angle u where the table is in equilibrium. The springs each have a stiffness of k = 200 N>m and are unstretched when u = 90. Neglect the mass of the links.
250 mm 150 mm
A 250 mm u
C u B
k
E
D
150 mm
1064
1134. If a 10-kg load I is placed on the pan, determine the position x of the 0.75-kg block H for equilibrium. The scale is in balance when the weight and the load are not on the scale.
50 mm 100 mm 100 mm A 100 mm B I C F D E
x
H
1065
1135. Determine the angles u for equilibrium of the 200-lb cylinder and investigate the stability of each position. The spring has a stiffness of k = 300 lb>ft and an unstretched length of 0.75 ft.
E
k
C
1.5 ft B 3 ft
A
u
u D
1066
*1136. Determine the angles u for equilibrium of the 50-kg cylinder and investigate the stability of each position. The spring is uncompressed when u = 60.
A 1m 1m
u
B k 900 N/m
C
1067
1137. If the mechanism is in equilibrium when u = 30, determine the mass of the bar BC. The spring has a stiffness of k = 2 kN>m and is uncompressed when u = 0. Neglect the mass of the links.
A u 450 mm
k 600 mm H
2 kN/m F
B
u D C
1068
1138. The uniform rod OA weighs 20 lb, and when the rod is in the vertical position, the spring is unstretched. Determine the position u for equilibrium. Investigate the stability at the equilibrium position.
3 ft A u O
1 ft
k
2 lb/in.
1139. The uniform link AB has a mass of 3 kg and is pin connected at both of its ends. The rod BD, having negligible weight, passes through a swivel block at C. If the spring has a stiffness of k = 100 N>m and is unstretched when u = 0, determine the angle u for equilibrium and investigate the stability at the equilibrium position. Neglect the size of the swivel block.
400 mm
D k 100 N/m
A u
C
400 mm B
1069
*1140. The truck has a mass of 20 Mg and a mass center at G. Determine the steepest grade u along which it can park without overturning and investigate the stability in this position.
G 3.5 m
1.5 m u 1.5 m
1141. The cylinder is made of two materials such that it has a mass of m and a center of gravity at point G. Show that when G lies above the centroid C of the cylinder, the equilibrium is unstable.
G C r a
1070
1142. The cap has a hemispherical bottom and a mass m. Determine the position h of the center of mass G so that the cup is in neutral equilibrium.
r
h G
1071
1143. Determine the height h of the cone in terms of the radius r of the hemisphere so that the assembly is in neutral equilibrium. Both the cone and the hemisphere are made from the same material.
h
r
1072
*1144. A homogeneous block rests on top of the cylindrical surface. Derive the relationship between the radius of the cylinder, r, and the dimension of the block, b, for stable equilibrium. Hint: Establish the potential energy function for a small angle u, i.e., approximate sin u L 0, and cos u L 1 - u2>2.
r
b b
1073
1145. The homogeneous cone has a conical cavity cut into it as shown. Determine the depth d of the cavity in terms of h so that the cone balances on the pivot and remains in neutral equilibrium.
h d r
1074
1146. The assembly shown consists of a semicylinder and a rectangular block. If the block weighs 8 lb and the semicylinder weighs 2 lb, investigate the stability when the assembly is resting in the equilibrium position. Set h = 4 in.
h 10 in. 4 in.
1075
1147. The 2-lb semicylinder supports the block which has a specific weight of g = 80 lb>ft3. Determine the height h of the block which will produce neutral equilibrium in the position shown.
h 10 in. 4 in.
1076
*1148. The assembly shown consists of a semicircular cylinder and a triangular prism. If the prism weighs 8 lb and the cylinder weighs 2 lb, investigate the stability when the assembly is resting in the equilibrium position.
6 in.
8 in. 4 in.
1077
1149. A conical hole is drilled into the bottom of the cylinder, and it is then supported on the fulcrum at A. Determine the minimum distance d in order for it to remain in stable equilibrium.
A d h
r
1078
1079
1150. The punch press consists of the ram R, connecting rod AB, and a flywheel. If a torque of M = 50 N # m is applied to the flywheel, determine the force F applied at the ram to hold the rod in the position u = 60.
B 0.1 m u
0.4 m R
F
M A
1080
1151. The uniform rod has a weight W. Determine the angle u for equilibrium. The spring is uncompressed when u = 90. Neglect the weight of the rollers.
B
L
k
A
u
1081
*1152. The uniform links AB and BC each weigh 2 lb and the cylinder weighs 20 lb. Determine the horizontal force P required to hold the mechanism at u = 45. The spring has an unstretched length of 6 in.
B
10 in.
10 in.
u A
k = 2 lb/in.
C
P
1082
1153. The spring attached to the mechanism has an unstretched length when u = 90. Determine the position u for equilibrium and investigate the stability of the mechanism at this position. Disk A is pin connected to the frame at B and has a weight of 20 lb.
C 1.25 ft u k u 1.25 ft A B u 16 lb/ft u
1083
1154. Determine the force P that must be applied to the cord wrapped around the drum at C which is necessary to lift the bucket having a mass m. Note that as the bucket is lifted, the pulley rolls on a cord that winds up on shaft B and unwinds from shaft A.
a
b B A C
c
P
1084
1155. The uniform bar AB weighs 100 lb. If both springs DE and BC are unstretched when u = 90, determine the angle u for equilibrium using the principle of potential energy. Investigate the stability at the equilibrium position. Both springs always remain in the horizontal position due to the roller guides at C and E.
k E u A 2 lb/in. D
B
k
4 lb/in. C
4 ft
2 ft
1085
*1156. The uniform rod AB has a weight of 10 lb. If the spring DC is unstretched when u = 0, determine the angle u for equilibrium using the principle of virtual work. The spring always remains in the horizontal position due to the roller guide at D.
k D 50 lb/ft u 1 ft A C
B
2 ft
1086
1157. Solve Prob. 1156 using the principle of potential energy. Investigate the stability of the rod when it is in the equilibrium position.
2 ft k D 50 lb/ft u 1 ft A C
B
1087
1158. Determine the height h of block B so that the rod is in neutral equilibrium. The springs are unstretched when the rod is in the vertical position. The block has a weight W.
B
h k k
l A
1088
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