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Unformatted text preview: APPENDIX The Fundamentals of Engineering (FE) exam is given semiannually by
the National Council of Engineering Examiners (NCEE) and is one of
the requirements for obtaining a Professional Engineering License. A
portion of this exam contains problems in dynamics, and this appendix
provides a review of the subject matter most often asked on this exam.
Before solving any of the problems, you should review the sections
indicated in each chapter in order to become familiar with the boldfaced
definitions and the procedures used to solve the various types of
problems. Also, review the example problems in these sections. The following problems are arranged in the same sequence as the
topics in each chapter. Besides helping as preparation for the FE exam,
these problems also provide additional examples for general practice of
the subject matter. Partial solutions and answers to all the problems are
given at the back of this appendix. Review for the Fundamentals of
Engineering Examination 455 456  APPENDIX D Review for the Fundamentals of Engineering Examination Chapter 12—»Review Sections 12.1, 12.44215,
12.8—12.9 Dl. The position of a particle is s = (0.5t3 + 4t) ft,
where t is in seconds. Determine the velocity and the
acceleration of the particle when r = 3 s. D2. After traveling a distance of 100 m, a particle
reaches a velocity of 30 m / s, starting from rest. Determine
its constant acceleration. 03. A particle moves in a straight line such that
s = (12t3 + 2t2 + 3t) m, wheretis in seconds. Determine
the velocity and acceleration of the particle when t = 2 s. D4. A particle moves along a straight line such that
a = (412 — 2) m/sz, wheretis in secondsWhent = 0, the
particle is located 2 m to the left of the origin, and when
t = 2 s, it is 20 m to the left of the origin. Determine the
position of the particle when t = 4 s. DS. Determine the speed at which the basketball at A
must be thrown at the angle of 30" so that it makes it to
the basket at B. Prob. [1—5 D6. A particle moves with curvilinear motion in the
x—y plane such that the y component of motion is
described by the equation )2 = (713)m, where t is in
seconds. If the particle starts from rest at the origin when
t = 0, and maintains a constant acceleration in the x
direction of 12 m/sz, determine the particle’s speed when
t = 2 s. D7. Water is sprayed at an angle of 90° from the slope
at 20 m/s. Determine the range R. UH 1' 20 “1’8 Prob. D~7 D8. An automobile is traveling with a constant speed
along a horizontal circular curve that has a radius of
p = 250 m. If the magnitude of acceleration is a =15 m/sz,
determine the speed at which the automobile is traveling. D9. A boat is traveling along a circular path having a
radius of 30 In. Determine the magnitude of the boat‘s
acceleration if at a given instant the boat’s speed is
v = 6 m/s and the rate of increase in speed is i) = 2 m/sz. D10. A train travels along a horizontal circular curve
that has a radius of 600 m. If the speed of the train is
uniformly increased from 40 km/h to 60 km/h in 5 5,
determine the magnitude of the acceleration at the instant
the speed of the train is 50 km / h. Dll. At a given instant, the automobile has a speed of
25 m/s and an acceleration of 3 m/s2 acting in the
direction shown. Determine the radius of curvature of the
path and the rate of increase of the automobile’s speed. n Prob. D—ll D12. At the instant shown, cars A and B are traveling
at the speeds shown. If B is accelerating at 1200 km/h2
while A maintains a constant speed, determine the
velocity and acceleration of A with respect to B. Prob. DlZ D13. Determine the speed of point P on the cable in
order to lift the platform at 2 m/s. Prob. D13 REVIEW PROBLEMS . 457 Chapter 13—Review Sections 13.1—13.5 D14. The effective weight of a man in an elevator
varies between 130 lb and 170 1b while he is riding in the
elevator. When the elevator is at rest the man weighs
153 lb. Determine how fast the elevator car can accelerate,
going up and going down. D15. Neglecting friction and the mass of the pulley and
cord, determine the acceleration at which the 4kg block
B will descend. What is the tension in the cord? Block A
has a mass of 2 kg. Prob. D~15 D16. The blocks are suspended over a pulley by a rope.
Neglecting the mass of the rope and the pulley, determine
the acceleration of both blocks and the tension in the rope. 25 kg Prob. D—lﬁ 458  APPENDiX D Review for the Fundamentals of Engineering Examination D17. Block B rests upon a smooth surface. If the D19. Determine the maximum speed that the jeep can
coefficients of static and kinetic friction between A and travel over the crest of the hill and not lose contact with
B are [LS = 0.4 and ,uk = 0.3, respectively, determine the the road. acceleration of each block if P = 6 lb. 20 lb Prob. Dl‘) D20. A pilot weighs 150 lb and is traveling at a constant
speed of 120 ft/s. Determine the normal force he exerts
on the seat of the plane when he is upside down at A. D18. The block rests at a distance of 2 m from the center The 10013 has a radius 0f curvature Of 400 ft of the platform. If the coefficient of static friction between the block and the platform is ,us = 0.3, determine the maximum speed which the block can attain before it begins to slip. Assume the angular motion of the disk is slowly increasing. Prob. D—IS Prob. D—ZO D21. The sports car is traveling along a 30° banked road
having a radius of curvature of p = 500 ft. If the coeffi
cient of static friction between the tires and the road is
[1'5 = 0.2, determine the maximum safe speed for travel so
no slipping occurs. Neglect the size of the car. Prob. DZl D22. The 5lb pendulum bob B is released from rest
when 0 = 0°. Determine the tension in string BC imme
diately after it is released and when the pendulum reach
es point D, where 0 = 90°. Prob. D—22 REVIEW PROBLEMS . 459 Chapter 14—Review All Sections D23. A 15 0001b freight car is pulled along a horizon
tal track. If the car starts from rest and attains a velocity
of 40 ft / s after traveling a distance of 300 ft, determine the
total work done on the car by the towing force in this dis tance if the rolling frictional force between the car and
track is 80 lb. D24. The 20lb block resting on the 30° inclined plane
is acted upon by a 40lb force. If the block’s initial velocity
is 5 ft / s down the plane, determine its velocity after it has
traveled 10 ft down the plane. The coefficient of kinetic
friction between the block and the plane is Mk = 0.2. Prob. D—24 D25. The 3—kg block is subjected to the action of the two
forces shown. If the block starts from rest, determine the
distance it has moved when it attains a velocity of
10 m / s. The coefficient of kinetic friction between the block
and the surface is Mk = 0.2. Prob. D—ZS 460 . APPENDIX D Review for the Fundamentals of Engineering Examination D26. The 6lb ball is to be fired from rest using a spring D28. The 5—lb collar is released from rest at A and having a stiffness of k = 401b/ft. Determine how far the travels along the frictionless guide. Determine the speed
spring must be compressed so that when the ball reaches of the collar when it strikes the stop B. The spring has
a height of 8 ft it has a velocity of 6 ft / s. an unstretched length of 0.5 ft. Mfr/Si @ 8ft “' : k = 401b/ft Prob. D—28 l’rob. D—26 D29. The 2—kg collar is given a downward velocity of 4 m / s
when it is at A. If the spring has an unstretched length of 1 m and a stiffness of k = 30 N/m, determine the velocity of the
D27. The 2kg pendulum bob is released from rest when collar at s = 1 m. it is at A. Determine the speed of the bob and the tension in
the cord when the bob passes through its lowest position, B. Prob. D—27 Prob. D~29 Chapter 1 5——Review Sections 15.1—15.4 D30. A 30ton engine exerts a constant horizontal force
of 40(103) lb on a train having three cars that have a total
weight of 250 tons. If the rolling resistance is 10 lb per ton
for both the engine and cars, determine how long it takes
to increase the speed of the train from 20 ft /s to 30 ft /s. What is the driving force exerted by the engine wheels on
the tracks? D31. A 5kg block is moving up a 30° inclined plane with
an initial velocity of 3 m /s If the coefficient of kinetic friction
between the block and the plane is ,uk = 0.3, determine how
long a 100N horizontal force must act on the block in order
to increase the velocity of the block to 10 m/s up the plane. D32. The 101b block A attains a velocity of 1 ft/s in
5 seconds, starting from rest. Determine the tension in the
cord and the coefficient of kinetic friction between block A and the horizontal plane. Neglect the weight of the pul—
ley. Block B has a weight of 8 1b. Prob. D—32 D33. Determine the velocity of each block 10 seconds
after the blocks are released from rest. Neglect the mass
of the pulleys. Prob. D—33 REVIEW PROBLEMS . 461 D34. The two blocks have a coefficient of restitution of
e = 0.5. If the surface is smooth, determine the velocity of
each block after impact. UA=2mls Prob. D—34 D35. A 6—kg disk A has an initial velocity of
(vA)1 = 20 m/s and strikes headon disk B that has a mass
of 24 kg and is originally at rest. If the collision is perfectly
elastic, determine the speed of each disk after the collision
and the impulse which disk A imparts to disk B. D36. Blocks A and B weigh 5 lb and 10 lb, respectively.
After striking block B, A slides 2 in. to the right, and B
slides 3 in. to the right. If the coefficient of kinetic friction
between the blocks and the surface is IJ~k = 0.2, determine
the coefficient of restitution between the blocks. Block B
is originally at rest. VA
—> Prob. D36 D37. Disk A weighs 2 lb and is sliding on the smooth
horizontal plane with a velocity of 3 ft /s. Disk B weighs
11 1b and is initially at rest. If after the impact A has a velocity
of 1 ft /s, directed along the positive x axis, determine the
speed of disk B after impact. y Prob. D—37 462  APPENDIX D Review for the Fundamentals of Engineering Examination Chapter 16—Review Sections 16.3, 16.5—16.7 D38. If gear A is rotating clockwise with an angular
velocity of wA = 3 rad/s, determine the angular velocities
of gears B and C. Gear B is one unit, having radii of 2 in.
and 5 in. (0A = 3 rad/s /. Prob. D—38 D39. The spin drier of a washing machine has a
constant angular acceleration of 2rev/s2, starting from
rest. Determine how many turns it makes in 10 seconds
and its angular velocity when t = 5 s. D40. Starting from rest, point P on the cord has a
constant acceleration of 20 ft/sz. Determine the angular
acceleration and angular velocity of the disk after it has
completed 10 revolutions. How many revolutions will the
disk turn after it has completed 10 revolutions and P
continues to move downward for 4 seconds longer? Prob. D—40 D41. The center of the wheel has a velocity of 3 m/s.
At the same time, it is slipping and has a clockwise angular
velocity of w = 2 rad/s. Determine the velocity of point
A at the instant shown. Prob.» 0—41 D42. A cord is wrapped around the inner core of the
gear and it is pulled with a constant velocity of 2 ft /s.
Determine the velocity of the center of the gear, C. Prob. D—42 REVIEW PROBLEMS . 463 D43. The center of the wheel is moving to the right D45. Determine the angular velocity of link AB at the
with a speed of 2 m/s. If no slipping occurs at the ground, instant shown. A, determine the velocity of point B at the instant shown. Prob. D—4S Prob. D—43 D44. If the velocity of the slider block at B is 2 ft / s to D46. When the slider block C is in the position shown,
the left, compute the velocity of the block at A and the the link AB has a clockwise angular velocity of 2 rad /s.
angular velocity of the rod at the instant shown. Determine the velocity of block C at this instant. Prob. D—44 Prob. D—46 464  APPENDIX D Review for the Fundamentals of Engineering Examination D47. The center of the pulley is being lifted vertically
with an acceleration of 3 m/sz, and at the instant shown
its velocity is 2 m/s. Determine the accelerations of points
A and B. Assume that the rope does not slip on the
pulley’s surface. (13:3 m/s2
1} Prob. D—47 D48. At a given instant, the slider block A has the velocity
and deceleration shown. Determine the acceleration of
block B and the angular acceleration of the link at
this instant. 17A=6ftf§A GA32ﬁ/32 Prob. D—48 Chapter 17—Review All Sections D49. The 35001b car has a center of mass located at
G. Determine the normal reactions of both front and both
rear wheels on the road and the acceleration of the car
if it is rolling freely down the incline. Neglect the weight
of the wheels. Prob. D—49 D50. The 20lb link AB is pinned to a moving frame at
A and held in a vertical position by means of a string BC
which can support a maximum tension of 10 lb. Determine
the maximum acceleration of the link without breaking the
string. What are the corresponding components of reaction
at the pin A? i——3ft Prob. D—50 DSl. The 50~lb triangular plate is released from rest.
Determine the initial angular acceleration of the plate
and the horizontal and vertical components of reaction
at B.The moment of inertia of the plate about the pinned
axis 13 is I3 = 2.30 slug'ftz. A Prob. DaSl D52. The 20—kg slender rod is pinned at 0. Determine
the reaction at 0 just after the cable is cut. Prob. D—52 D53. The 20kg wheel has a radius of gyration of
KG = 0.8 m. Determine the angular acceleration of the
wheel if no slipping occurs. Prob. D—53 REVlEW PROBLEMS . 465 D54. The 15~kg wheel has a wire wrapped around its
inner hub and is released from rest on the inclined plane,
for which the coefficient of kinetic friction is Mk = 0.1. If
the centroidal radius of gyration of the wheel is k0 = 0.8 m,
determine the angular acceleration of the wheel. Prob. D—54 DSS. The 2kg gear is at rest on the surface of a gear
rack. If the rack is suddenly given an acceleration of
5 m/sz, determine the initial angular acceleration of the
gear. The radius of gyration of the gear is kg = 0.3 m. 5 m/s2 Prob. DSS 466 . APPENDIX D Review for the Fundamentals of Engineering Examination Solutions and Answers Dl. v = £3— : 1.5t2 + 4,=3 = 17.5 ft/s Ans.
dv
a = E = 3t,=3 = 9ft/s2 Ans. D2. 30)2 = (0)2 + 2a(100 — 0)
a = 4.5 m/s2 Ans. d
D3. v = d—: = 36:2 + 41‘ + 3L:2 = 155 m/s Ans.
a = % = 72: + 4L:2 = 148 m/s2 Ans. D4. v = “4:2 — 2) dt 4
n=~:3—2t+C1 3
43
S: 3t —2t+C1 dt
1
S=§t4—12+C1[+C2 t=0,s= —2,C2= —2
I: 2,s = —20, C1 = —9.67
t=4,s =28.7m Ans. D5. i’S 2 50 + 1101
10=0+vAcos30°t 1 2
+T S = SO ‘1' Dot ‘1' Eact 1
3 = 1.5 + DA sin 30°; + E(—9.81):2
= 0933. DA = 12.4 m/s Ans. D6. v = 00 + act
12X = 0 + 12(2) = 24 m/s
dy
0. = :1? = 21t2I,:2 = 84 m/s
1) = (24)2 + (84)2 = 87.4 m/s Ans. D7. (i)s=so+vot R<4>=o+2o<2>3 (+T) s = so + Dot + 551612 3 4 1 2
— — = + — +— —.
R(5) 0 20(5) 2( 981)t t = 5.105
R = 76.5 m Ans. DS. a, = 0
712
an — a — 1.5 — ﬁ’ 0 — 19.4 m/s Ans.
D9. a, = 2m/s2
2 6 2
a" = v; = % = 1.20 m/s2
0 = (2)2 + (1.20)2 = 2.33 m/s2 Ans.
Av 60 — 40
Dl. 2—: — = 14400k h2
0 a’ At [(5 — 0)/3600] m/
v2 (50)2
= — = = 4167 k h2
a" p 0.6 m/
a = (14.4)2 + (4.167)2103 = 15.0(103)km/h2
Ans.
Dll. a, = 3 cos 40° = 2.30 m/s2 Ans.
v2 . (25V
a,,=—;381n40°= ,p=324m Ans. p D12. VA 2 VB + VA/B
—20 cos 45°i + 20 sin 45°j = 65i + vA/B
vA/B = —79.14i + 14.14j vA/B = (—79.14)2 + (14.14)2 = 80.4 km/h Ans.
3A 2 a3 + aA/B
(20V (20V o+
0‘1 c0845 1 01 aA/B = 1628i + 2828j aA/B = \ /(1628)2 + (2828)2 = 326(103) km/h2 Ans. D13. 4sA + sp :1 sin 45°j = 1200i + aA/B Up = —4vA = — (—2) = 8m/s Ans.
153
D14. +T EFy = may; 170 — 153 = an
a = 3.58 ft/s2 T Ans.
153 ,
+ iEFy = may; 153 — 130 = Ea a' = 4.84 ft/s2 i Ans. D15. Block B:
+l EFy = may; 4(9.81) — T = 4a
Block A:
¢ EFX = max; T = 2a
T = 13.1 N, a = 6.54 m/s2 Ans. D16. D17. D18. D19. D20. D21. D22. Block A: +1 sz = may;15(9.81) — T = —15a
Block B: +1213 = may; 25(9.81) — T = 25a
a = 2.45 m/sz, T = 184N Ans. Blocks A and B: ibEF— 70 z— = 2
max;6 322nm 2.76ft/s Check if slipping occurs between A and B. 20
i, = . _ _
2F. max, 6 F— 32—2(2. .76) F = 4.29 lb < 0.4(20) — 8 lb aA = a3 = 2.76 m/s2 Ans. 2 U2 2F” = m%; (0.3)m(9.81) = m3
7) = 2.43 m/s Ans. 2
v
+ _ _
‘LEFn— man; m(32. 2)— m<2—50>
v = 89.7 ft/s Ans.
2 N _15_0((120)2>
P ‘ 32.2 400
1) JEFf — man;N sin30°+ 0.2Ncos30°= m— 500
+T EFb = 0;
NC cos 30° — 0.2 NC sin 30° — m(32.2) = 0
v = 119 ft/s Ans. +1 2F" = man; 150 +
Np = 17.7 lb Ans. At B: $ = = _
2F” mamT (3:2)<05 In the general position, 8
N8
N +/'EF,, =man ;T—55in6 5
+ — 5 — —
\ 2F,— ma,; cos 6— 32 2 90
J 64.4c050d0 = Iv vdv
0 0 TA
m‘
V
A [CM
V v = 11.3 ft/s,
When 0 = 90°;
T = 15 lb Ans. D23. D24. D25. D26. D27. D28. D29. SOLUTIONS AND ANSWERS  467 T1 ‘1’ 2U1_2 : T2
15000
+ _
0 U1_ 2 80(300)= $(3—22 >(4 40)
U1_2 = 397(103) ftlb Ans. T1 '1' 21112 2 T2 %<3—2202)(5)2 + 40(10)— . 1 20 2
u o + o : — —
(o 2)(20 cos 30 )(10) 2000 5m 30 > 2 (32.2)” v = 39.0 ft/s Ans. +1213 = may;Nb + 100 sin 20° — 10 — 3(9.81) =
Nb = 5.23 N
T1 + 2U1_2 = T2 1
+ (100 cos 20°)d — 0.2(5.23)d = E(3)(10)2
d = 1.61 m Ans. n+1w=n+w
0 +— 2x(40)( x2): l(—6—>(6)2 + 6(8) 2 32.2
x = 1.60 ft Ans. TA '1‘ VA = TB ‘1‘ VB
1
0 + 2(9.81)(1.5) = 5(2)(63)2 + 0 123: 5.42 m/s Ans. (5.42)2
+1215 = man; T — 2(9.81) = 2 15
T = 58.9N Ans.
TA ‘1' VA = TB + VB
0 + %(4)(2.5 — 0.5)2 + 5(25)
_ l i 2 l _ 2
— 2(32.2)03 + 2(4)(1 0.5) v3 = 16.0 ft/s Ans. T1+W=T2+112 §<2><4>2 + §(30>(2 — 02 = 112x202 — 219.800) + 2
v = 5.26 m/s Ans. %(30)(\/5 — 1)2 468 . D'30. D31. D32. D33. D34. Three cars: D35.
j5va + Edet = mvz
250(2000) 3
' _ 250(2000) 30
7 32.2 ( )
t = 4.145 Ans.
Engine:
i>mvl + 2)th = mvz
30(2000) 20 + F 414 10 30 414
322 ( ) (  ) ( )(  ) D36.
_ 30(2000) 30
_ 32.2 ( )
F = 4800 lb Ans.
+'\ EFy = 0;
Nb — 5(9.81) cos 30° — 100 sin 30° = 0
Nb = 92.48 N
+/'m’U1 + Edet 2 mvz
5(3) + (100 cos 30°)t — 5(9.81) sin 30°(t) —
0.3(92.48)t = 5(10)
t = 1.025 Ans.
Block B:
(+l)mv1+ det = mm
8
0 + 8(5) — T(5) = EU) D37.
T = 7.95 lb Ans.
Block A:
($)mv1+ det = mvz
10
0 + 795(5) — Mk(10)(5 )= 372(1)
Mk‘ — 0.789 Ans
2sA + 53 = l
20A = “’03
+1, m(7}A)1 ‘l' IF dt : m(vA)2
0 + 10(9.81)(10) — 2T(10) = 10(vA)2 D38.
+i m(vB)1 + IF d‘ = m(vB)2
0 + 50(9.81)(10) — T(10) = 50(vB)2
T = 70.1 N
(71A)2 = —42.0 m/s = 42.0 m/sT Ans.
(vB)2 = 84.1 m/sl Ans.
$2M]: Emvz; 5(2) — 2(5): 5(1) 22) + 2(2).)2 D'39'
i)e_(vB)2— (”A)2 J05_(B B)2 (”/02
(DA)! — (UB)1 2 — (—5)
(vA)2 — —1 m/s=1m/s<— Ans. (v3)2 = 2.5 m/s—> Ans. APPENDIX D Review for the Fundamentals of Engineering Examination $Emv1=6(20) + 0:6(11A)2+ 24(7)B)2
_ (7)3)2— (UA)2 _ (”B)2 _ (vA)2
33:? — ; 1 —
(71/01 — (UB)1 20 _ 0
(1),,» = —12m/s = 12 m/s<— Ans.
(v3); = 8 m/s—> Ans.
Disk A:
mv1+ det = mvz;
6(20) — [F dt = —6(12)
de1‘ = 192Ns Ans. After collision: T] + EU1_2 = T2 X32002) "2) —02(5)<122>= 0
(pm— — 1.46511/5 %(3—:°2)< W>02<10>(1—Z)= 0
(v3)2— — 1.794ft/s Emvl = 2mm 5 5 10
__ + _
322(1)")1 0 322(1'Jr465) 322( (UA)1 = 5.054
(vB)2 — (DA)2 =
(UA)1 ‘ (UB)1 1 794) 1.794 — 1.465
5.054 — 0 e = = 0.0652 Ans. 2"1(vx)1 = 2"1(vx)2
2 11 0 + 0 ‘ 322(1) + 322(1)”)2
(v3.92 = —0.1818ft/s
Em(vy 1 = 2m('vy )2 2 11 )
(3) + 0— _ O + *(vBy)2 32.2 32. 2 (03y)2 = 0.545 ft/S (1)3)2 : (—0.1818)2 + (0.545)2 = 0.575 ft/s Ans. (03(5) = 3(4)
103 = 2.40 rad/5’3 Ans. wC(3) = 240(2)
wC = 1.60 rad/s) Ans. 1
0 = 00 + wot + Eact2 1
— 2 1 2:
2( )( 0) 100 rev Ans. w = we + act
= 0 + 2(5) = 10 rev/s 9=0+0+ A ns. D40. D41. D42. D43. D44. D45. a 2
a = 7’ = 20 = 10 rad/s2 Ans. 10 rev = 2077 rad (02 = wﬁ + 2ac(6 — 90)
wz = 0 + 2(10)(207T — 0)
w = 354 rad/s Ans. 1 2
0 = 60 + (Hot + adv! 1
9 = 0 + 35.4(4) + 2(10)(4)2 = 222 rad 0 — Q = 35.3 rev. — 27r Ans. VA : VC ‘1‘ (A) X rA/C
vAi = 3i + (—2k) >< (—0.5j)
12A = 2m/s—+ Ans. vA=vB+erA/B 2i = 0 + (~wk) >< (1.5j)
a) = 1.33 rad/s2
VC:VB+erC/B vCi = 0 + (—1.33k) X (1j)
vc = 1.33 ft/s—&...
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