Course Hero has millions of student submitted documents similar to the one

below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Mechanics Engineering - Dynamics
Chapter 18
Problem 18-1 At a given instant the body of mass m has an angular velocity and its mass center has a velocity vG. Show that its kinetic energy can be represented as T = 1/2 IIC 2, where IIC is the moment of inertia of the body computed about the instantaneous axis of zero velocity, located a distance rGIC from the mass center as shown. Solution: T 1 2 m vG 2 1 m 2 rGIC 1 2
2
IG
2
where vG =
rGIC
T
1 2 IG 2 IG
2
T
1 2 m rGIC 2 1 2 IIC 2
However m(rGIC)2 + IG = IIC
T
Problem 18-2 The wheel is made from a thin ring of mass mring and two slender rods each of mass mrod. If the torsional spring attached to the wheel's center has stiffness k, so that the torque on the center of the wheel is M = k , determine the maximum angular velocity of the wheel if it is rotated two revolutions and then released from rest. Given: mring mrod k r 2 5 kg 2 kg Nm rad
0.5 m
Solution: IO IO T1 2 1 2 mrod ( 2r) 12
2
mring r
2
1.583 kg m U12 T2
591
Engineering Mechanics - Dynamics
Chapter 18
0
0
4
k d
1 2 IO 2
k 4 IO
14.1
rad s
Problem 18-3 At the instant shown, the disk of weight W has counterclockwise angular velocity center has velocity v. Determine the kinetic energy of the disk at this instant. Given: W 30 lb 5 rad s ft s when its
v r g
20 2 ft
32.2
ft s
2
Solution: 1 1 W 2 r 2 2 g 1 W 2 v 2 g
T
2
T
210 ft lb
*Problem 18-4 The uniform rectangular plate has weight W. If the plate is pinned at A and has an angular velocity , determine the kinetic energy of the plate. Given: W 30 lb 3 a b Solution: T 1 2 m vG 2 1 2 IG 2
592
rad s
2 ft 1 ft
Engineering Mechanics - Dynamics
Chapter 18
T T
1 W 2 g 6.99 ft lb
b
2
a
2
2
2
1 1 W 2 12 g
b
2
a
2
2
Problem 18-5 At the instant shown, link AB has angular velocity AB. If each link is considered as a uniform slender bar with weight density , determine the total kinetic energy of the system. Given:
AB
rad s lb 0.5 in 2 45 deg
a b c
3 in 4 in 5 in
Solution: Guesses rad s in s
g
BC
1
CD
1
rad s T 1 lb ft
vGx Given 0 0
1
vGy
1
in s
0 a 0
0 0
BC
b 0 0 b 2 0
0 0
CD
c cos c sin 0 0
AB
0 0
AB 3
0 a 0
0 0
BC 3
vGx vGy 0
3
0
2 BC
T
1 2
a 3
2 AB
1 2
b 12
1 2 b vGx 2
vGy
2
1 2
c 3
2 CD
593
Engineering Mechanics - Dynamics
Chapter 18
BC CD BC CD
vGx vGy T
Find
BC
CD vGx vGy T
1.5 1.697
rad s T
vGx vGy
0.5 0.25
ft s
0.0188 ft lb
Problem 18-6 Determine the kinetic energy of the system of three links. Links AB and CD each have weight W1, and link BC has weight W2. Given: W1 W2
AB
10 lb 20 lb 5 rad s
rAB rBC rCD g
1 ft 2 ft 1 ft 32.2 ft s
2
Solution:
BC
0
rad s
2
rAB
CD AB
rCD 1 W1 rCD 2 g 3
2 2 CD
T
1 W1 2 g 10.4 ft lb
rAB 3
2 AB
1 W2 2 g
AB rAB
2
T
Problem 18-7 The mechanism consists of two rods, AB and BC, which have weights W1 and W2, respectively, and a block at C of weight W3. Determine the kinetic energy of the system at the instant shown, when the block is moving at speed vC.
594
Engineering Mechanics - Dynamics
Chapter 18
Given: W1 W2 W3 rAB rBC vC g 3 10 lb 20 lb 4 lb 2 ft 4 ft ft s ft s Solution:
BC 2
32.2
0
rad s
2
vC
AB
rAB 1 W2 2 vC 2 g 1 W3 2 vC 2 g
T
1 W1 2 g
rAB 3
2 AB
T
3.82 lb ft
*Problem 18-8 The bar of weight W is pinned at its center O and connected to a torsional spring. The spring has a stiffness k, so that the torque developed is M = k If the bar is released from rest when it is vertical at = 90 , determine its angular velocity at the instant = 0 Use the prinicple of work and energy. Given: W k a g Solution:
0
10 lb 5 lb ft rad
1 ft 32.2 ft s
2
90 deg
f
0 deg
595
Engineering Mechanics - Dynamics
Chapter 18
Guess 1 2 k 0 2
1
rad s 1 2 k f 2 1 W ( 2a) 2 g 12 10.9 rad s
2 2
Given
Find
Problem 18-9 A force P is applied to the cable which causes the reel of mass M to turn since it is resting on the two rollers A and B of the dispenser. Determine the angular velocity of the reel after it has made two revolutions starting from rest. Neglect the mass of the rollers and the mass of the cable. The radius of gyration of the reel about its center axis is kG. Given: P M kG 20 N 175 kg 0.42 m 30 deg ri r0 a Solution: 0 P2 2 ri 8 P ri M kG Problem 18-10 The rotary screen S is used to wash limestone. When empty it has a mass M1 and a radius of gyration kG. Rotation is achieved by applying a torque M about the drive wheel A. If no slipping occurs at A and the supporting wheel at B is free to roll, determine the angular velocity of the screen after it has rotated n revolutions. Neglect the mass of A and B.
596
250 mm 500 mm 400 mm
1 2 2 M kG 2 2.02 rad s
2
Engineering Mechanics - Dynamics
Chapter 18
Unit Used: rev Given: M1 kG M rS rA n Solution: rS rA 1 2 2 M1 kG 2 800 kg 1.75 m 280 N m 2m 0.3 m 5 rev 2 rad
M
n
2M rS n rA M1 kG
2
6.92
rad s
Problem 18-11 A yo-yo has weight W and radius of gyration kO. If it is released from rest, determine how far it must descend in order to attain angular velocity . Neglect the mass of the string and assume that the string is wound around the central peg such that the mean radius at which it unravels is r. Given: W kO 0.3 lb 0.06 ft rad s
70
r g
0.02 ft 32.2 ft s
2
597
Engineering Mechanics - Dynamics
Chapter 18
Solution: 1 W 2 g kO 2g
2 2 2
0
Wh
2
r
1 W 2 kO 2 g
2
r h
h
0.304 ft
*Problem 18-12 The soap-box car has weight Wc including the passenger but excluding its four wheels. Each wheel has weight Ww, radius r, and radius of gyration k, computed about an axis passing through the wheel's axle. Determine the car's speed after it has traveled a distance d starting from rest. The wheels roll without slipping. Neglect air resistance. Given: Wc Ww r k d 110 lb 5 lb 0.5 ft 0.3 ft 100 ft 30 deg g 32.2 ft s Solution: Wc 4Ww d sin 1 Wc 4Ww 2 v g 2 g k r
2 2
1 Ww 2 4 k 2 g ft s
v r
2
v
2 Wc Wc
4Ww d sin 4Ww 4Ww
v
55.2
2
598
Engineering Mechanics - Dynamics
Chapter 18
Problem 18-13 The pendulum of the Charpy impact machine has mass M and radius of gyration kA. If it is released from rest when = 0 determine its angular velocity just before it strikes the specimen S, = 90 . Given: M kA d g 50 kg 1.75 m 1.25 m 9.81 m s Solution: 0 Mgd 1 2 2 M kA 2 2 2.83 rad s
2
2g d
2
kA
2
2
Problem 18-14 The pulley of mass Mp has a radius of gyration about O of kO. If a motor M supplies a force to the cable of P = a (b ce dx), where x is the amount of cable wound up, determine the speed of the crate of mass Mc when it has been hoisted a distance h starting from rest. Neglect the mass of the cable and assume the cable does not slip on the pulley. Given: Mp Mc kO r h Solution: Guesses vc 1 m s 10 kg 50 kg 0.21 m 0.3 m 2m a b c d 800 N 3 2 1 m
599
Engineering Mechanics - Dynamics
Chapter 18
h
Given
0
ab
ce
dx
dx
1 2
2 vc Mp kO
2
1 2
r
Mc vc
2
Mc g h
vc
Find vc
vc
9.419
m s
Problem 18-15 The uniform pipe has a mass M and radius of gyration about the z axis of kG. If the worker pushes on it with a horizontal force F, applied perpendicular to the pipe, determine the pipe's angular velocity when it has rotated through angle about the z axis, starting from rest. Assume the pipe does not swing. Units Used: Mg Given: M kG F Solution: 0 Fl 1 2 M kG
2 2
10 kg
3
16 Mg 2.7 m 50 N r l
90 deg 0.75 m 3m
1 M
MFl kG
0.0636
rad s
*Problem 18-16 The slender rod of mass mrod is subjected to the force and couple moment. When it is in the position shown it has angular velocity 1. Determine its angular velocity at the instant it has rotated downward 90. The force is always applied perpendicular to the axis of the rod. Motion occurs in the vertical plane.
600
Engineering Mechanics - Dynamics
Chapter 18
Given: mrod
1
4 kg 6 rad s
F M a g
15 N 40 N m 3m 9.81 m s
2
Solution:
Guess
2
1
rad s
Given
2 2 1 2 2 2
1 mrod a 2 3 Find
Fa
2 8.25
mrod g rad s
a 2
M
1 mrod a 2 2 3
2
2
2
Problem 18-17 The slender rod of mass M is subjected to the force and couple moment. When the rod is in the position shown it has angular velocity 1. Determine its angular velocity at the instant it has rotated 360. The force is always applied perpendicular to the axis of the rod and motion occurs in the vertical plane. Given: mrod
1
4 kg 6 rad s
M a g
40 N m 3m 9.81 m s
2
F Solution:
15 N
Guess
2
1
rad s
601
Engineering Mechanics - Dynamics
Chapter 18
Given 1 mrod a 2 3
2 2 1
F a2
M2
1 mrod a 2 rad s 3
2 2 2
2
Find
2
2
11.2
Problem 18-18 The elevator car E has mass mE and the counterweight C has mass mC. If a motor turns the driving sheave A with constant torque M, determine the speed of the elevator when it has ascended a distance d starting from rest. Each sheave A and B has mass mS and radius of gyration k about its mass center or pinned axis. Neglect the mass of the cable and assume the cable does not slip on the sheaves.
Units Used: Given: mE mC mS M Solution: d r
Mg
1000 kg
1.80 Mg 2.30 Mg 150 kg 100 N m
d r k g
10 m 0.35 m 0.2 m 9.81 m s
2
M
mE d r
mC g d
1 2
mE
mC
2mS
k r
2
2
v
2
2M v
mE mC
mC g d 2mS k r
2
v
4.973
m s
mE
2
Problem 18-19 The elevator car E has mass mE and the counterweight C has mass mC. If a motor turns the driving sheave A with torque a 2 + b, determine the speed of the elevator when it has ascended a distance d starting from rest. Each sheave A and B has mass mS and radius of gyration k about its mass center or pinned axis. Neglect the mass of the cable and assume the cable does not slip on the sheaves.
602
Engineering Mechanics - Dynamics
Chapter 18
Units Used: Mg Given: mE mC mS a b d r k g Solution: Guess v
d
1000 kg
1.80 Mg 2.30 Mg 150 kg 0.06 N m 7.5 N m 12 m 0.35 m 0.2 m 9.81 m s
2
1
m s
2
Given
r
a
0
2
bd
mE m s
mC g d
1 2
mE
mC
2mS
k r
2
v
2
v
Find ( v)
v
5.343
*Problem 18-20 The wheel has a mass M1 and a radius of gyration kO. A motor supplies a torque M = (a + b), about the drive shaft at O. Determine the speed of the loading car, which has a mass M2, after it travels a distance s = d. Initially the car is at rest when s = 0 and = 0. Neglect the mass of the attached cable and the mass of the car's wheels.
603
Engineering Mechanics - Dynamics
Chapter 18
Given: M1 M2 d a b r kO 100 kg 300 kg 4m 40 N m 900 N m 0.3 m 0.2 m 30 deg Solution: Guess v
d r
1
m s
2
Given
0
a
b d m s
1 2
M2 v
2
1 2
M1 kO
2 v
r
M2 g d sin
v
Find ( v)
v
7.49
Problem 18-21 The gear has a weight W and a radius of gyration kG. If the spring is unstretched when the torque M is applied, determine the gear's angular velocity after its mass center G has moved to the left a distance d. Given: W M ro ri d k 15 lb 6 lb ft 0.5 ft 0.4 ft 2 ft 3 lb ft
604
Engineering Mechanics - Dynamics
Chapter 18
kG Solution: Guess
0.375 ft
1
rad s 1 W 2 g
2
Given
M
d ro
ro
1 W 2 g
2 2 kG
1 2
k
ri ro
ro
2
d
Find
7.08
rad s
Problem 18-22 The disk of mass md is originally at rest, and the spring holds it in equilibrium. A couple moment M is then applied to the disk as shown. Determine its angular velocity at the instant its mass center G has moved distance d down along the inclined plane. The disk rolls without slipping. Given: md M d k 20 kg 30 N m 0.8 m 150 N m Guess 1 rad s md g sin 0.654 m r g 30 deg 0.2 m 9.81 m s
2
Solution:
Initial stretch in the spring md g sin k
k d0 d0
d0 Given M d r
md g d sin
k 2
d
d0
2
d0
2
1 2
md
r
2
1 1 2 2
md r
2
2
605
Engineering Mechanics - Dynamics
Chapter 18
Find
11.0
rad s
Problem 18-23 The disk of mass md is originally at rest, and the spring holds it in equilibrium. A couple moment M is then applied to the disk as shown. Determine how far the center of mass of the disk travels down along the incline, measured from the equilibrium position, before it stops. The disk rolls without slipping. Given: md M k 20 kg 30 N m 150 N m
30 deg r g 0.2 m 9.81 m s Solution:
2
Guess
d
3m
Initial stretch in the spring md g sin k d r
k d0 d0
md g sin 0.654 m
d0
Given
M
md g d sin Find ( d) d
k 2
d
d0
2
d0
2
0
d
2m
*Problem 18-24 The linkage consists of two rods AB and CD each of weight W1 and bar AD of weight W2. When = 0, rod AB is rotating with angular velocity
O.
If rod CD is subjected to a couple moment M
606
Engineering Mechanics - Dynamics
Chapter 18
and bar AD is subjected to a horizontal force P as shown, determine Given: W1 W2 8 lb 10 lb rad s
1
AB
at the instant
=
1.
a b
2 ft 3 ft
0
2
90 deg 15 lb ft
P
20 lb ft s
2
M
g
32.2
Solution: U P a sin 1 1 M 1 rad s 2W1 a 2 1 cos
1
W2 a 1
cos
1
Guess Given 1 2 W 1 a2 g 3
2
2 0
1 W2 2 g
a 0
2
U
1 2
2
W 1 a2 g 3
2
1 W2 2 g
a
2
Find
5.739
rad s
Problem 18-25 The linkage consists of two rods AB and CD each of weight W1 and bar AD of weight W2. When = 0, rod AB is rotating with angular velocity O. If rod CD is subjected to a couple moment M and bar AD is subjected to a horizontal force P as shown, determine AB at the instant = 1.
607
Engineering Mechanics - Dynamics
Chapter 18
Given: W1 W2 8 lb 10 lb rad s
1
a b
2 ft 3 ft
0
2
45 deg 20 lb
M
15 lb ft ft s
2
P
g
32.2
Solution: U P a sin 1 1 M 1 rad s
2 0
2W1
a 2
1
cos
1
W2 a 1
cos
1
Guess
Given
1 2
2
W 1 a2 g 3
1 W2 2 rad s g
a 0
2
U
1 2
2
W 1 a2 g 3
2
1 W2 2 g
a
2
Find
5.916
Problem 18-26 The spool has weight W and radius of gyration kG. A horizontal force P is applied to a cable wrapped around its inner core. If the spool is originally at rest, determine its angular velocity after the mass center G has moved distance d to the left. The spool rolls without slipping. Neglect the mass of the cable. Given: W 500 lb d 6 ft
608
Engineering Mechanics - Dynamics
Chapter 18
kG P g
1.75 ft 15 lb 32.2 ft s
2
ri ro
0.8 ft 2.4 ft
Solution:
Guess ro ro ri
1
rad s 1 W 2 g ro
2
Given
P
d
1 W 2 2 kG 2 g
Find
1.324
rad s
Problem 18-27 The double pulley consists of two parts that are attached to one another. It has a weight Wp and a centroidal radius of gyration kO and is turning with an angular velocity clockwise. Determine the kinetic energy of the system. Assume that neither cable slips on the pulley. Given: WP WA WB 50 lb 20 lb 30 lb 20 rad s r1 r2 kO 0.5 ft 1 ft 0.6 ft
Solution: KE 1 2 I 2 1 2 WA vA 2 1 2 WB vB 2 1 WA 2 g
2
KE KE
1 WP 2 2 kO 2 g 283 ft lb
r2
1 WB 2 g
r1
2
609
Engineering Mechanics - Dynamics
Chapter 18
*Problem 18-28 The system consists of disk A of weight WA, slender rod BC of weight WBC, and smooth collar C of weight WC. If the disk rolls without slipping, determine the velocity of the collar at the instant the rod becomes horizontal, i.e. = 0 The system is released from rest when = Given: WA WBC WC
0
0.
20 lb 4 lb 1 lb 45 deg
L r g
3 ft 0.8 ft 32.2 ft s
2
Solution: Guess Given L WBC cos 2 1 WC 2 vC 2 g ft s
2 1 WBC L 2 g 3
vC
1
ft s
0
WC L cos
vC L
2
0
vC
Find vC
vC
13.3
Problem 18-29 The cement bucket of weight W1 is hoisted using a motor that supplies a torque M to the axle of the wheel. If the wheel has a weight W2 and a radius of gyration about O of kO, determine the speed of the bucket when it has been hoisted a distance h starting from rest.
610
Engineering Mechanics - Dynamics
Chapter 18
Given: W1 W2 M kO h r Solution: Guess v h r 1 ft s 1 W1 2 g v
2
1500 lb 115 lb 2000 lb ft 0.95 ft 10 ft 1.25 ft
Given
M
1 W2 2 6.41 g ft s
kO
2 v
2
r
W1 h
v
Find ( v)
v
Problem 18-30 The assembly consists of two slender rods each of weight Wr and a disk of weight Wd. If the spring is unstretched when = 1 and the assembly is released from rest at this position, determine the angular velocity of rod AB at the instant = 0. The disk rolls without slipping. Given: Wr Wd
1
15 lb 20 lb 45 deg 4 lb ft
k L r Solution: Guess
3 ft 1 ft
1
rad s
611
Engineering Mechanics - Dynamics
Chapter 18
Given
2Wr
L 2
sin 1
1 2 rad s
k 2L
2L cos
2 1
2
1 1 Wr 2 L 2 3 g
2
Find
4.284
Problem 18-31 The uniform door has mass M and can be treated as a thin plate having the dimensions If shown. it is connected to a torsional spring at A, which has stiffness k, determine the required initial twist of the spring in radians so that the door has an angular velocity when it closes at = 0 after being opened at = 90 and released from rest. Hint: For a torsional spring M = k where k is the stiffness and is the angle of twist. Given: M k 20 kg 80 Nm rad rad s a b 0.8 m 0.1 m
12 P
c
2m
0N
Solution: Guess Given
0
0
1 rad
k d
0 90 deg
1 1 2 3
Ma
2 2
0
Find
0
0
1.659 rad
*Problem 18-32 The uniform slender bar has a mass m and a length L. It is subjected to a uniform distributed load w0 which is always directed perpendicular to the axis of the bar. If it is released from the position shown, determine its angular velocity at the instant it has rotated 90. Solve the problem for rotation in (a) the horizontal plane, and (b) the vertical plane.
612
Engineering Mechanics - Dynamics
Chapter 18
Solution:
2 0
L
x w0 dx d
0
1 2 L w0 4
(a)
1 2 L w0 4
1 1 2 mL 2 3
2
3 w0 2m
(b)
1 2 L w0 4
1 1 2 mL 2 3
2
mg
L 2
3 w0 2m
3g L
Problem 18-33 A ball of mass m and radius r is cast onto the horizontal surface such that it rolls without slipping. Determine the minimum speed vG of its mass center G so that it rolls completely around the loop of radius R + r without leaving the track.
Solution: v R
2 2
mg
m
v
2
gR
2
1 2 2 mr 2 5 1 2 vG 5
vG r
1 2 m vG 2 1 gR 5
m g2R 1 gR 2
1 2 2 mr 2 5
v r vG
1 2 mv 2 3 3 gR 7
1 2 vG 2
2g R
613
Engineering Mechanics - Dynamics
Chapter 18
Problem 18-34 The beam has weight W and is being raised to a vertical position by pulling very slowly on its bottom end A. If the cord fails when = 1 and the beam is essentially at rest, determine the speed of A at the instant cord BC becomes vertical. Neglect friction and the mass of the cords, and treat the beam as a slender rod. Given: W
1
1500 lb 60 deg 13 ft 12 ft 7 ft 32.2 ft s
2
L h a g Solution: W
L sin 1 2
h 2
a
1 W 2 vA 2 g h 2 a ft s
vA
2g
L sin 1 2
vA
14.2
Problem 18-35 The pendulum of the Charpy impact machine has mass M and radius of gyration kA. If it is released from rest when = 0, determine its angular velocity just before it strikes the specimen S, = 90 using the conservation of energy equation. Given: M kA d g 50 kg 1.75 m 1.25 m 9.81 m s
2
614
Engineering Mechanics - Dynamics
Chapter 18
Solution: 0 Mgd 0 rad s 1 2 2 M kA 2 2 2g d
2
kA
2
2
2.83
*Problem 18-36 The soap-box car has weight Wc including the passenger but excluding its four wheels. Each wheel has weight Ww radius r, and radius of gyration k, computed about an axis passing through the wheel's axle. Determine the car's speed after it has traveled distance d starting from rest. The wheels roll without slipping. Neglect air resistance. Solve using conservation of energy. Given: Wc Ww r k Solution: 1 Wc 4Ww 2 v 2 g g k r
2
110 lb 5 lb 0.5 ft 0.3 ft
d
100 ft 30 deg ft s
2
g
32.2
0
Wc
4Ww d sin
0
1 Ww 2 4 k 2 g ft s
v r
2
v
2 Wc Wc
4Ww d sin 4Ww 4Ww
v
55.2
2
Problem 18-37 The assembly consists of two slender rods each of weight Wr and a disk of weight Wd. If the spring is unstretched when = 1 and the assembly is released from rest at this position, determine the angular velocity of rod AB at the instant Solve using the conservation of energy. = 0. The disk rolls without slipping.
615
Engineering Mechanics - Dynamics
Chapter 18
Given: Wr Wd
1
15 lb 20 lb 45 deg
k L r
4
lb ft
3 ft 1 ft rad s
Solution: Given
Guess
1
0
2Wr
L sin 1 2
2
1 1 Wr 2 L 2 3 g rad s
2
1 k 2L 2
2L cos
2 1
Find
4.284
Problem 18-38 A yo-yo has weight W and radius of gyration kO. If it is released from rest, determine how far it must descend in order to attain angular velocity . Neglect the mass of the string and assume that the string is wound around the central peg such that the mean radius at which it unravels is r. Solve using the conservation of energy. Given: W kO 0.3 lb 0.06 ft 70 r g Solution: 0 Ws
2
rad s
0.02 ft 32.2 ft s
2
1 W 2 g kO 2g
2
r
2
1 W 2 kO 2 g
2
0
r s
2
s
0.304 ft
616
Engineering Mechanics - Dynamics
Chapter 18
Problem 18-39 The beam has weight W and is being raised to a vertical position by pulling very slowly on its bottom end A. If the cord fails when = 1 and the beam is essentially at rest, determine the speed of A at the instant cord BC becomes vertical. Neglect friction and the mass of the cords, and treat the beam as a slender rod. Solve using the conservation of energy. Given: W
1
1500 lb 60 deg 13 ft 12 ft 7 ft 32.2 ft s
2
L h a g Solution:
0
W
L sin 1 2 L sin 1 2
1 W 2 vA 2 g h 2 a
W
h 2
a
vA
2g
vA
14.2
ft s
*Problem 18-40 The system consists of disk A of weight WA, slender rod BC of weight WBC, and smooth collar C of weight WC. If the disk rolls without slipping, determine the velocity of the collar at the instant the rod becomes horizontal, i.e. = 0 The system is released from rest when = 0. Solve using the conservation of energy. Given: WA WBC 20 lb 4 lb L r 3 ft 0.8 ft
617
Engineering Mechanics - Dynamics
Chapter 18
WC
0
1 lb 45 deg
g
32.2
ft s
2
Solution: Guess Given L 2 1 WC
0 2 2 1 WBC L
vC
1
ft s
0
WBC
cos
0
WC L cos ft s
2
g
vC
vC L
2
2
g
3
0
vC
Find vC
vC
13.3
Problem 18-41 The spool has mass mS and radius of gyration kO. If block A of mass mA is released from rest, determine the distance the block must fall in order for the spool to have angular velocity . Also, what is the tension in the cord while the block is in motion? Neglect the mass of the cord. Given: ms mA 5 Solution: Guesses Given 0 0 1 2 ms kO 1 2
2 2
50 kg 20 kg rad s
ri ro kO
0.2 m 0.3 m 0.280 m
g
9.81
m s
2
d
1m
T
1N
1 2
mA ri
2
mA g d
0 d T
0
Td
mA ri
2
mA g d
Find ( d T)
d
0.301 m
T
163 N
618
Engineering Mechanics - Dynamics
Chapter 18
Problem 18-42 When slender bar AB of mass M is horizontal it is at rest and the spring is unstretched. Determine the stiffness k of the spring so that the motion of the bar is momentarily stopped when it has rotated downward 90. Given: M a b g Solution: 0 0 0 1 2 k (a b)
2
10 kg 1.5 m 1.5 m 9.81 m s
2
a
2
2
b
Mg
a 2 N m
k (a
Mga b)
2
a
2
2
k
42.8
b
Problem 18-43 The disk of weight W is rotating about pin A in the vertical plane with an angular velocity when = 0. Determine its angular velocity at the instant shown, 90 deg. Also, compute the horizontal and vertical components of reaction at A at this instant. Given: W
1
15 lb 2 rad
s 90 deg 0.5 ft 32.2 ft s
2
r g
Solution: Guesses Ax 1 lb Ay 1 lb
2
1
rad s
1
rad s
2
619
Engineering Mechanics - Dynamics
Chapter 18
Given
1 3 W 2 r 2 2 g Ax W g
2 1 2
Wr
1 3 W 2 r 2 2 g Ay W W g
2 2
r 2
r
Wr
3 W 2 g
r
2
Ax Ay
2
Find A x A y
2
42.9
rad s
2 2
9.48
rad s
Ax Ay
20.9 5.0
lb
*Problem 18-44 The door is made from one piece, whose ends move along the horizontal and vertical tracks. If the door is in the open position = 0 , and then released, determine the speed at which its end A strikes the stop at C. Assume the door is a thin plate of weight W having width c. Given: W a b c g Solution: Guesses vA 1 ft s 1 rad s 180 lb 3 ft 5 ft 10 ft 32.2 ft s
2
Given 1 W (a 2 g b) 12
2
0
0
b
a 2
b
2
2
W
a 2
b
vA
b
vA
Find
vA
6.378
rad s
vA
31.9
ft s
620
Engineering Mechanics - Dynamics
Chapter 18
Problem 18-45 The overhead door BC is pushed slightly from its open position and then rotates downward about the pin at A. Determine its angular velocity just before its end B strikes the floor. Assume the door is a thin plate having a mass M and length l. Neglect the mass of the supporting frame AB and AC. Given: M l h Solution: 180 kg 6m 5m
2
d
h
l 2
2
Guess
1
rad s 1 Ml 2 12
2
Given
Mgd
Md
2
2
Find
2.03
rad s
Problem 18-46 The cylinder of weight W1 is attached to the slender rod of weight W2 which is pinned at point A. At the instant = the rod has an angular velocity 0 as shown. Determine the angle
f
to which the rod swings before it momentarily stops.
Given: W1 W2
0 0
80 lb 10 lb 1 rad s
a b l g
1 ft 2 ft 5 ft 32.2 ft s
2
30 deg
621
Engineering Mechanics - Dynamics
Chapter 18
Solution: W1 1 a 2 g 4 2 b 2 W1
f
IA
b 12 l 2
2
W1 g
l
b 2
2
W 2 l2 g 3
W1 l d
W2 W2
Guess Given
1 deg W1 W2 d cos
0
1 2 IA 0 2
f f
W1
W2 d cos
f
f
Find
39.3 deg
Problem 18-47 The compound disk pulley consists of a hub and attached outer rim. If it has mass mP and radius of gyration kG, determine the speed of block A after A descends distance d from rest. Blocks A and B each have a mass mb. Neglect the mass of the cords. Given: mp kG d Solution: Guess Given 0 0 1 2 mb vA 2 ri 1 mb vA 2 ro
2
3 kg 45 mm 0.2 m
ri ro g
30 mm 100 mm 9.81 m s
2
mb
2 kg
vA
1
m s
1 2 mp kG 2
vA ro
2
mb g d
mb g
ri ro
d
vA
Find vA
vA
1.404
m s
622
Engineering Mechanics - Dynamics
Chapter 18
*Problem 18-48 The semicircular segment of mass M is released from rest in the position shown. Determine the velocity of point A when it has rotated counterclockwise 90. Assume that the segment rolls without slipping on the surface. The moment of inertia about its mass center is IG. Given: M IG Solution: Guesses 1 rad s 1 2 M vG 2 vG 1 m s M g( d r) m s vG d 2 r 15 kg 0.25 kg m
2
r d
0.15 m 0.4 m
Given
Mgd
1 2 IG 2 12.4 rad s
vG
Find
vG d 0 2 d 2 0
vG
0.62
2.48 vA 2.48 0.00 m s vA 3.50 m s
vA
0
Problem 18-49 The uniform stone (rectangular block) of weight W is being turned over on its side by pulling the vertical cable slowly upward until the stone begins to tip. If it then falls freely ( T = 0) from an essentially balanced at-rest position, determine the speed at which the corner A strikes the pad at B. The stone does not slip at its corner C as it falls. Given: W a b g Solution: Guess 1 rad s
623
150 lb 0.5 ft 2 ft 32.2 ft s
2
Engineering Mechanics - Dynamics
Chapter 18
Given
W
a
2
b
2
2 b vA
1 W a b 2 g 3 11.9 ft s
2
2
2
W
a 2
Find
vA
Problem 18-50 The assembly consists of pulley A of mass mA and pulley B of mass mB. If a block of mass mb is suspended from the cord, determine the block's speed after it descends a distance d starting from rest. Neglect the mass of the cord and treat the pulleys as thin disks. No slipping occurs. Given: mA mB mb d r R g 3 kg 10 kg 2 kg 0.5 m 30 mm 100 mm 9.81 m s Solution: Given 1 mA r 2 2 Find vb
2 2
Guess
vb
1
m s
0
0
vb r vb
2
1 mB R 2 2 1.519 m s
2
vb R
2
1 2 mb vb 2
mb g d
vb
Problem 18-51 A uniform ladder having weight W is released from rest when it is in the vertical position. If it is allowed to fall freely, determine the angle at which the bottom end A starts to lift off the ground. For the calculation, assume the ladder to be a slender rod and neglect friction at A.
624
Engineering Mechanics - Dynamics
Chapter 18
Given: W L g Solution: (a) The rod will rotate around point A until it loses contact with the horizontal constraint ( A x = 0). We will find this point first Guesses
1
30 lb 10 ft 32.2 ft s
2
30 deg
1
1
rad s
1
1
rad s
2
Given L 2 1 1 W 2 L 2 3 g 1 W 2 L 3 g
2 L 1 1 2 1
0
W
W
L cos 2
1
W
L sin 1 2 L cos 2
1
1
2
sin 1
0
1 1 1
Find
1
1
1
1
48.19 deg
1
1.794
rad s
1
3.6
rad s
2
(b) Now the rod moves without any horizontal constraint. If we look for the point at which it loses contact with the floor (Ay = 0 ) we will find that this condition never occurs.
*Problem 18-52 The slender rod AB of weight W is attached to a spring BC which, has unstretched length L. If the rod is released from rest when = 1,determine its angular velocity at the instant = 2. Given: W 25 lb
625
Engineering Mechanics - Dynamics
Chapter 18
L k
1 2
4 ft 5 lb ft 30 deg 90 deg 32.2 ft s
2
g
Solution: Guess Given 0 L sin 1 W 2 1 2 kL 2 21 cos
1
1
rad s
2
1
2
1 W L L 2 W sin 2 2 g 2 3 2 1 2 kL 2 1 cos 2 1 2
Find
1.178
rad s
Problem 18-53 The slender rod AB of weight w is attached to a spring BC which has an unstretched length L. If the rod is released from rest when = 1, determine the angular velocity of the rod the instant the spring becomes unstretched. Given: W L k 25 lb 4 ft lb 5 ft g
1
30 deg 32.2 ft s
2
Solution: When the spring is unstretched rad s
626
2
120 deg
Guess
1
Engineering Mechanics - Dynamics
Chapter 18
Given L sin 1 2 1 2 kL 2
2
0
W
21
cos
1
1
1 W L L 2 W sin 2 2 g 2 3 2 1 2 kL 2 1 cos 2 1 2
2
Find
2.817
rad s
Problem 18-54 A chain that has a negligible mass is draped over the sprocket which has mass ms and radius of gyration kO. If block A of mass mA is released from rest in the position s = s1, determine the angular velocity of the sprocket at the instant s = s2. Given: ms kO mA s1 s2 r g 2 kg 50 mm 4 kg 1m 2m 0.1 m 9.81 m s Solution: Guess 1 rad s
2
Given 0 mA g s1 1 mA r 2
2
1 2 2 ms kO 2 41.8 rad s
mA g s2
Find
627
Engineering Mechanics - Dynamics
Chapter 18
Problem 18-55 A chain that has a mass density is draped over the sprocket which has mass ms and radius of gyration kO. If block A of mass mA is released from rest in the position s = s1, determine the angular velocity of the sprocket at the instant s = s2. When released there is an equal amount of chain on each side. Neglect the portion of the chain that wraps over the sprocket. Given: ms kO mA g 2 kg 50 mm 4 kg 9.81 m s Solution: Guess T1 V1 Given T1 V1 T2 0 mA g s1 1 mA r 2 mA g s2 V1 T2
2 2
s1 s2 r
1m 2m 0.1 m 0.8 kg m
1Nm 1Nm
T2 V2
1Nm 1Nm
10
rad s
2 s1 g
s1 2 1 2 2s1 2s1 r
2
1 2 2 ms kO 2 s2 g V2 s2 2
V2 T1 T1 V1 T2 V2
s2 g
2s1 2
s2
Find T1 V1 T2 V 2
39.3
rad s
628
Engineering Mechanics - Dynamics
Chapter 18
*Problem 18-56 Pulley A has weight WA and centroidal radius of gyration kB. Determine the speed of the crate C of weight WC at the instant s = s2. Initially, the crate is released from rest when s = s1. The pulley at P "rolls" downward on the cord without slipping. For the calculation, neglect the mass of this pulley and the cord as it unwinds from the inner and outer hubs of pulley A. Given: WA WC kB s1 Solution: Guess Given WC s1 1 WA 2 2 kB 2 g 1 WC 2 vC 2 g rad s WC s2 vC rA 2 rB 1 rad s vC 1 ft s 30 lb 20 lb 0.6 ft 5 ft rA rB rP s2 0.4 ft 0.8 ft rB 2 10 ft rA
vC
Find
vC
18.9
vC
11.3
ft s
Problem 18-57 The assembly consists of two bars of weight W1 which are pinconnected to the two disks of weight W2. If the bars are released from rest at = 0 , determine their angular velocities at the instant = 0. Assume the disks roll without slipping. Given: W1 W2 r l
0
8 lb 10 lb 0.5 ft 3 ft 60 deg
629
Engineering Mechanics - Dynamics
Chapter 18
Solution: Guess 1 rad s 2 1 W1 2 g l 3 rad s
2 2
Given
2W1
l sin 0 2
Find
5.28
Problem 18-58 The assembly consists of two bars of weight W1 which are pin-connected to the two disks of weight W2. If the bars are released from rest at = 1, determine their angular velocities at the instant = 2. Assume the disks roll without slipping. Given: W1 W2
1 2
8 lb 10 lb 60 deg 30 deg 0.5 ft 3 ft
r l
Solution: Guesses 1 rad s vA 1 ft s
Given l 2W1 sin 1 2 vA l sin 2
630
l 2W1 sin 2 2
1 W1 2 2 g
l 3
2
2
1 3 W2 2 2 r 2 2 g
vA r
2
Engineering Mechanics - Dynamics
Chapter 18
vA
Find
vA
vA
3.32
ft s
2.21
rad s
Problem 18-59 The end A of the garage door AB travels along the horizontal track, and the end of member BC is attached to a spring at C. If the spring is originally unstretched, determine the stiffness k so that when the door falls downward from rest in the position shown, it will have zero angular velocity the moment it closes, i.e., when it and BC become vertical. Neglect the mass of member BC and assume the door is a thin plate having weight W and a width and height of length L. There is a similar connection and spring on the other side of the door. Given: W L a Solution: Guess Given
2
200 lb 12 ft 1 ft
b
2 ft 15 deg
k
1
lb ft
d
1 ft
b
L 2
2
d
2
2d
L cos 2
2
0
W
L 2
1 L 2 k 2 2
b
d
k d
Find ( k d)
d
4.535 ft
k
100.0
lb ft
631

**Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.**

Below is a small sample set of documents:

Western Michigan - ME - 2580

Engineering Mechanics - DynamicsChapter 22Problem 22-1 When a load of weight W1 is suspended from a spring, the spring is stretched a distance d. Determine the natural frequency and the period of vibration for a load of weight W2 attached to the

Western Michigan - ME - 2580

Engineering Mechanics - DynamicsChapter 21aIyz02m a h2h( ay)h a y dy 2 21 mah 6Iyz1 mah 6Problem 21-7 Determine by direct integration the product of inertia Ixy for the homogeneous prism. The density of the material is . Expre

Western Michigan - ME - 2580

Engineering Mechanics - DynamicsChapter 20Problem 20-1 The ladder of the fire truck rotates around the z axis with angular velocity 1 which is increasing at rate 1. At the same instant it is rotating upwards at the constant rate 2. Determine the

Western Michigan - ME - 2580

Engineering Mechanics - DynamicsChapter 19Problem 19-1 The rigid body (slab) has a mass m and is rotating with an angular velocity about an axis passing through the fixed point O. Show that the momenta of all the particles composing the body can

Delaware - CHEM - 119H

EXPERIMENT 6 SPECTROPHOTOMETRIC DETERMINATION OF A TWO-COMPONENT MIXTURE 07F Reference: Harris, Chap 19; Lecture notes; R. E. Kitson, Analyt. Chem., 22, 664 (1950). Remarkably few real samples actually contain only one component for any type of analy

Delaware - CHEM - 119H

1 EXPERIMENT 7 ANALYSIS OF VINEGARS 07F Quality control is important for any commercial product. However, it is not likely that two batches of a product are exactly the same. The analyses of solutions are easier than the analyses of solids because a

University of Florida - STA - 3032

STA3032 Section 7347 Quiz 4.Name: UFID#:1. Following are the residual plots for the regression of y on x1 and x2 : What is the most appropriate conclusion? (3 points)Normal Probability Plot of the Residuals5qResidual vs Fitted values5q4

Delaware - CHEM - 119H

1 EXPERIMENT 8 CHARACTERIZATION OF A WEAK ACID 07F It is apparent from tables in the texts for CHEM 111 and for CHEM 119 that all weak acids do not have identical ionization constants. Therefore, acids may be identified by their ionization constants

Delaware - CHEM - 119H

EXPERIMENT 9 CONDUCTOMETRIC TITRATIONS 07F Measurement of the conductivity of a solution (generally aqueous) is perhaps the easiest way to determine whether the compound is a strong electrolyte, a weak electrolyte, or a nonelectrolyte. The conductivi

Delaware - CHEM - 119H

EXPERIMENT 10 TITRATION OF ASPIRIN 07F The Web is replete with procedures for the determination of aspirin, acetylsalicylic acid {ASA}, in commercial tablets as experiments for quantitative analysis. The procedure that appeared to be most common was

Pittsburgh - MATH - 0230

1. Determine the volume of the solid formed by rotating the region bounded by y = 2x and y = x2 for 0 x 2 about the x-axis. 2 04x - x4 dx = 32 15 - ln 4 52. Determine the volume of the solid formed by rotating the region bounded by the x-axi

Pittsburgh - MATH - 0220

o w cm w j x u n o n j x z Ah $v f o z m z j w cm z Ah v d u c3FDiD 9DD9 u 9Hq3ii0 9F39Hf } if f u s fF } u 759cD9av f f e e u u x d| ~ V0ph v Du cc3eq s u f u A97 s 3ica9D } f e gf e d 9D ccBF

UCLA - MATH - 32B

yellowMATH 32B Final Exam LAST NAME FIRST NAME ID NO.March 20, 2008Your TA:To receive credit, you must write your answer in the space provided.DO NOT WRITE BELOW THIS LINE1 (20 pts) 2 (20 pts) 3 (20 pts) 4 (20 pts)5 (20 pts) 6 (20 pts)

UCLA - MATH - 32B

S E C T I O N 16.3Triple Integrals(ET Section 15.3)919This gives the following inequalities of S: S : 0 z 4, 0 x The upper surface z = 4 - y 2 can be described by y = obtain the following iterated integral:1-z 4 4 - z. We4 - z,

UCLA - MATH - 32B

1032C H A P T E R 16M U LTI P L E I N T E G R AT I O N3 9-x 2(ET CHAPTER 15)15. ExpressSOLUTION-3 0f (x, y) d y d x as an iterated integral in the order d x d y.The limits of integration correspond to the inequalities describing the d

UCLA - MATH - 32B

Chapter Review Exercises1271=- =y 0y x 2 + t 2 - 2z 2 + t 2 + z 2 - 2x 2 dt 2 2 2 5/2 0(x + t + z )(x + t + z )x 2 - 2t 2 + z 2 dt = 2 2 2 5/2y3/2 (x 2 + y 2 + z 2 )= Q(x, y, z)The last integral can be verified by showing that y

UCLA - MATH - 32B

MATH 32B SECOND MAKE-UP MIDTERM 26, 2004MAYLAST NAME FIRST NAME ID NO.Please write clearly and legibly. There are four 20-point problems and four multiple choice problems worth 5 points each. To receive credit, you must circle your answer. DO

UCLA - MATH - 32B

MATH 32B SECOND MIDTERMMAY 26, 2004LAST NAME FIRST NAME ID NO.Please write clearly and legibly. There are four 20-point problems and four multiple choice problems worth 5 points each. To receive credit, you must circle your answer. DO NOT WRIT

UCLA - MATH - 32B

MATH 32B SECOND MIDTERMMAY 26, 2004LAST NAME FIRST NAME ID NO.Please write clearly and legibly. There are four 20-point problems and four multiple choice problems worth 5 points each. To receive credit, you must circle your answer. DO NOT WRIT

UCLA - MATH - 32B

1050C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)= e2r cosh2 (s) - e2r sinh2 (s) = e2r (cosh2 (s) - sinh2 (s) = e2r 42. Find a linear mapping (u, v) that maps the unit square to the parallelogram in the x y-plane spanned b

UCLA - MATH - 32B

1044C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)yD 2 xTherefore, W is described by W : x 2 + y 2 z 8 - (x 2 + y 2 ), (x, y) D Thus, M=8-(x 2 +y 2 )Dx 2 +y 2(x 2 + y 2 )1/2dz d x d yWe convert the integr

UCLA - MATH - 32B

S E C T I O N 18.3y x 2 + y2 = 9 D 3 xDivergence Theorem(ET Section 17.3)1255We convert the integral to polar coordinates: x = r cos ,S2y = r sin ,0 r 3,3 00 2 9r 2 r4 3 - 2 4 0 = 81F dS =2 0 032 9 - r 2 r dr d = 4

UCLA - MATH - 32B

S E C T I O N 17.5Surface Integrals of Vector Fields(ET Section 16.5)1141Using the Surface Integral over a Graph we have: Area(S) = 1 dS =D2 1 + gx + g 2 d A y(1)SIn parametrizing the surface by (x, y) = (x, y, g(x, y), (x, y) = D, w

UCLA - MATH - 32B

1160C H A P T E R 17L I N E A N D S U R FA C E I N T E G R A L S(ET CHAPTER 16)24. Calculate the flow rate through the upper hemisphere of the sphere x 2 + y 2 + z 2 = R 2 (z 0) for v as in Exercise 23.SOLUTIONWe use the parametrization,

UCLA - MATH - 32B

1148C H A P T E R 17L I N E A N D S U R FA C E I N T E G R A L S(ET CHAPTER 16)8. F = x, y, z ,SOLUTION1 part of sphere x 2 + y 2 + z 2 = 1, where z 23 , 2inward-pointing normalzy xWe parametrize S by the following parametriz

UCLA - MATH - 32B

S E C T I O N 17.5Surface Integrals of Vector Fields(ET Section 16.5)1163In Exercises 2930, a varying current i(t) flows through a long straight wire in the x y-plane as in Example 5. The i current produces a magnetic field B whose magnitude

UCLA - MATH - 32B

S E C T I O N 17.1Vector Fields(ET Section 16.1)106119. F = x, 0, zSOLUTIONThis vector field is shown in (A) (by process of elimination). x x 2 + y2 + z2 , y x 2 + y2 + z2 , z x 2 + y2 + z220. F =SOLUTIONThe unit radial vector field i

UCLA - MATH - 32B

LINE AND 17 SURFACE INTEGRALS17.1 Vector FieldsPreliminary Questions1. Which of the following is a unit vector field in the plane? (a) F = y, x (b) F = (c) F = y x 2 + y2 , x x 2 + y2(ET Section 16.1)y x , x 2 + y2 x 2 + y2SOLUTION(a) The

UCLA - MATH - 32B

986C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)1 If you had arranged the axes differently, you could have computed the answer as 0, 12 , 37 (depending on orientation). 4866. According to Coulomb's Law, the force between

UCLA - MATH - 32B

974C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)z =0 = 4 4The spherical inequalities for W are thus W : 0 2 , 0 We obtain the following integral: 1 dV = = =2 0 2 0 0 0 , 0 4 cos 4 /404 cos W2 sin

UCLA - MATH - 32B

S E C T I O N 16.4Integration in Polar, Cylindrical, and Spherical Coordinates(ET Section 15.4)96948. Calculate the volume of the sphere x 2 + y 2 + z 2 = a 2 , using both spherical and cylindrical coordinates.SOLUTIONSpherical coordinates

UCLA - MATH - 32B

1170C H A P T E R 17L I N E A N D S U R FA C E I N T E G R A L S(ET CHAPTER 16)Therefore the vector 4y, -x is tangent to the ellipses. Since F(x, y) is a scalar multiple of this vector, i.e, F(x, y) = 1 4y, -x , F is parallel to 4y, -x hence

UCLA - MATH - 32B

S E C T I O N 18.3Divergence Theorem(ET Section 17.3)1245(b) By Stokes' Theorem,CrF ds =Srcurl(F) dSBy part(a) we have m(r ) 1 F ds M(r ) r 2 Cr (2)We take the limit over the circles of radius r centered at Q, as r 0. As r

UCLA - MATH - 32B

1260C H A P T E R 18F U N D A M E N TA L TH E O R E M S O F V E C T O R A N A LY S I S(ET CHAPTER 17)25. Prove that div( f g) = 0.SOLUTIONWe compute the cross product: f g = f x , f y , f z gx , g y , gz = i fx gx j fy gy k fz gz= f

UCLA - MATH - 32B

1008C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)Therefore, D is defined by the inequalities 1u uv Since x = v+1 and y = v+1 , we havey 2, x3 y+x 6y = v+1 = v u x v+1uvandy+x =u u(v + 1) uv + = =u v+1 v+1 v

UCLA - MATH - 32B

S E C T I O N 16.5Change of Variables(ET Section 15.5)100325. WithSOLUTIONas in Example 3, use the Change of Variables Formula to compute the area of the image of [1, 4] [1, 4]. Let R represent the rectangle [1, 4] [1, 4]. We proceed as

UCLA - MATH - 32B

1000C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)(1, 0) = ( A 1 + C 0, B 1 + D 0) = ( A, B) = (1, 7) We substitute in (1) to obtain the following map: (u, v) = (u - 2v, 7u + 5v)A = 1,B=721. Let D be the paralle

UCLA - MATH - 32B

S E C T I O N 17.3Conservative Vector Fields(ET Section 16.3)1099h (z) = 0 Substituting in (2) we geth(z) = C (x, y, z) = x 2 yz + CSince only one potential function is needed, we choose the one corresponding to C = 0. That is, (x, y

UCLA - MATH - 32B

1064C H A P T E R 17L I N E A N D S U R FA C E I N T E G R A L S(ET CHAPTER 16)y Q = (c, d)P = (a, b) R = (c, b) D xSince x (x, y) = 0 in D, in particular x (x, b) = 0 for a x c. Therefore, for a x c we have (x, b) =x au

UCLA - MATH - 32B

1084C H A P T E R 17L I N E A N D S U R FA C E I N T E G R A L S(ET CHAPTER 16)40. Let C be the path from P to Q in Figure 17 that traces C1 , C2 , and C3 in the orientation indicated. Suppose thatCF ds = 5,C1F ds = 8,C3F ds = 8

UCLA - MATH - 32B

946C H A P T E R 16M U LTI P L E I N T E G R AT I O N3(ET CHAPTER 15)09-y 214.0SOLUTIONx 2 + y2 d x d yThe region D is defined by the following inequalities: 0 y 3,y0x 9 - y24 3 2 1 Dx 0 1 2 3 4We see that D is the qu

UCLA - MATH - 32B

930C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)The centroid of the tetrahedron is thus P = (1.5, 1, 2). 36. Find the centroid of the region described in Exercise 30. The region W is bounded by the cylinders z = 1 - y 2 an

UCLA - MATH - 32B

936C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)A5 = A4 C 5 =8 2 2 16 = 5.26 2 15 15The maximum value is A5 . Combining with (3) we conclude that the five-dimensional ball has the largest volume. By the closed formu

UCLA - MATH - 32B

S E C T I O N 16.4SOLUTIONIntegration in Polar, Cylindrical, and Spherical Coordinates(ET Section 15.4)941Step 1. Express W in cylindrical coordinates. W is bounded above by the plane z = x and below by z = 0, therefore 0 z x, in particula

UCLA - MATH - 32B

908C H A P T E R 16M U LTI P L E I N T E G R AT I O N1 0(ET CHAPTER 15)=13 y2 - 3 2dy =13 y3 1 25 1 13 1 y- - = =4 = 3 6 0 3 6 6 616. Integrate f (x, y, z) = z over the region W below the upper hemisphere of radius 3 as in Figure 12,

UCLA - MATH - 32B

S E C T I O N 16.2yDouble Integrals over More General Regions(ET Section 15.2)893y = x2D 1 xTherefore, ydA =1 0 0 x2Dy dy dx =0 2x2 1 1 2 y y=0dx =1 1 0 2x4 - 0 dx =1 1 x5 1 1 x4 dx = = 2 10 0 10 0(2)We compute the a

UCLA - MATH - 32B

S E C T I O N 16.5Change of Variables(ET Section 15.5)99116.5 Change of VariablesPreliminary Questions(ET Section 15.5)1. Which of the following maps is linear? (a) (uv, v) (b) (u + v, u)SOLUTION(c) (3, eu )(a) This map is not linea

UCLA - MATH - 32B

996C H A P T E R 16M U LTI P L E I N T E G R AT I O N(ET CHAPTER 15)The image of the horizontal line v = c is the set of the points (x, y) = (u, c) = (eu , eu+c ) x = eu , y = ec x y = ec x, x >0Since u can take any value, x can take any

UCLA - MATH - 32B

S E C T I O N 16.5Change of Variables(ET Section 15.5)1013Let1 (u, v) = ( Au + Cv, Bu + Dv). We ask that 1 (0, 1) = (1, 1), 1 (1, 0) = (1, -1)That is,1 (0, 1) = (C, D) = (1, 1) 1 (1, 0) = ( A, B) = (1, -1) C = 1, A = 1,D=1 B = -1

UCLA - MATH - 32B

S E C T I O N 17.4Parametrized Surfaces and Surface Integrals(ET Section 16.4)1117= (3 + sin v) cos u)i + (3 + sin v) sin u)j - (3 + sin v) cos v)k Hence, n(u, v) = (3 + sin v) 1 + cos2 v We obtain the following area: Area(S) = n du dv =2 0

UCLA - MATH - 32B

1094C H A P T E R 17L I N E A N D S U R FA C E I N T E G R A L S(ET CHAPTER 16)t1=t0y(t)x (t) + x(t)y (t) dt =t1 d t0dt(x(t)y(t) dtThe last equality follows from the Product Rule for differentiation. We now use the Fundamental Th

UCLA - MATH - 32B

S E C T I O N 17.5Surface Integrals of Vector Fields(ET Section 16.5)1155Step 1. Compute the tangent and normal vectors. We have, Tu = = u 3 - v, u + v, v 2 = 3u 2 , 1, 0 u u = u 3 - v, u + v, v 2 = -1, 1, 2v Tv = v v i 3u 2 -1 j 1 1 k 0

UCLA - MATH - 32B

1106C H A P T E R 17L I N E A N D S U R FA C E I N T E G R A L S(ET CHAPTER 16)Further Insights and Challenges25. The vector field F = (a) (b) (c) (d) x Show that F satisfies the cross-partials condition on D. Show that (x, y) = 1 ln(x 2 +

UCLA - MATH - 32B

1134C H A P T E R 17L I N E A N D S U R FA C E I N T E G R A L S(ET CHAPTER 16)43. Prove a famous result of Archimedes: The surface area of the portion of the sphere of radius r between two horizontal planes z = a and z = b is equal to the su

Michigan State University - WRA - 135

Jose l Rodriguez August 27, 2007What math classes did you take in highschool? In high school I took Algebra1, Geometry and Algebra 2 What is your major? My major is agriculture Goal for this semester for this class what do you expect from Ina and i

Cornell - ORIE - 350

ORIE 350 Homework #6 Due March 5, 2008 1. Camelrock Corporation reported the following relating to a lease transaction on its December 31, 2004 balance sheet (the company entered into the lease transaction on January 1, 2004): Leased Asset Gross Les

BU - STATS - MN 308

lJ rs.re"tepoir: *,{U*,?rL-onT\vlzLtou:,Jr , {| g M f (x,3) y= ' fl*r =pfr=",j)" _*L_Iur &v-rrji " TfuTJuu , c - i| . J- ffi ]X{ .*y*S - EI!_!LF, F*,(x,l , p-(r<r,f t,v*'! 4o=) 7ap ^ ,(r, y )^ '/ )," {xryfx ,\) >,,xd i'f{,1,)o

Georgia Tech - CIVIL - 6536

Pittsburgh - CHEM - 320

Chemistry 0320 - Spring 2008 Ch 15 - Benzene and AromaticityChapter 15: Benzene and AromaticityProblem sets to skip:.61 This chapter is about the aromaticity of benzene and electrophilic aromatic substitution.15-1 Nomenclaturebenzene aromatic c

Pittsburgh - CHEM - 320

Chemistry 0320 - Spring 2008 Ch 16 - Electrophilic Attack on Derivatives of BenzeneChapter 16: Electrophilic Attack on Derivatives of BenzeneSubstituents control regioselectivity.16-1 Activation or Deactivation by Substituents on a Benzene Ring

Pittsburgh - CHEM - 320

Chemistry 0320 - Spring 2008 Ch 23 - Ester Enolates and Claisen CondensationChapter 23: Ester Enolates and Claisen Condensation23-1 Claisen CondensationMechanismWrong match-up between the base and substrate:page 1Chemistry 0320 - Spring 20