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Problem 18.31 Points Band C lie in the x–yplane.
The yaxis is vertical. The center of mass of the 18-
kg arm BC is at the midpoint of the line from B
Solution: From the solution of Problem 18.30, the acceleration of
point Bis aB=−0.323i−0.149j(m/s2).IfαBC =0, the acceleration
of the center of mass Gof arm BC is
Problem 18.32 The radius of the 2-kg disk is R=
80 mm. Its moment of inertia is I=0.0064 kg-m2.It
rolls on the inclined surface. If the disk is released from
rest, what is the magnitude of the velocity of its center
two seconds later? (See Active Example 18.2).
30⬚
R
Solution: There are four unknowns (N, f, a, α), three dynamic
equations, and one constraint equation. We have
Problem 18.33 The radius of the 2-kg disk is R=
80 mm. Its moment of inertia is I=0.0064 kg-m2. What
minimum coefficient of static friction is necessary for the
disk to roll, instead of slip, on the inclined surface? (See
R
Problem 18.34 A thin ring and a homogeneous circu-
lar disk, each of mass mand radius R, are released from
rest on an inclined surface. Determine the ratio vring/vdisk
of the velocities of the their centers when they ave rolled
a distance D.
D
RR
D
490
Problem 18.35 The stepped disk weighs 40 lb and its
moment of inertia is I=0.2 slug-ft2. If the disk is
released from rest, how long does it take its center to
fall 3 ft? (Assume that the string remains vertical.)
4 in
Problem 18.36 The radius of the pulley is R=
100 mm and its moment of inertia is I=0.1 kg-
m2. The mass m=5 kg. The spring constant is k=
135 N/m. The system is released from rest with the
spring unstretched. At the instant when the mass has
fallen 0.2 m, determine (a) the angular acceleration of
R
=(0.1m)([5 kg][9.81 m/s2]−[135 N/m][0.2m])
492
Problem 18.37 The radius of the pulley is R=100 mm
and its moment of inertia is I=0.1 kg-m2. The mass
m=5 kg. The spring constant is k=135 N/m. The sys-
tem is released from rest with the spring unstretched.
What maximum distance does the mass fall before re-
bounding?
R
Problem 18.38 The mass of the disk is 45 kg and its
radius is R=0.3 m. The spring constant is k=60 N/m.
The disk is rolled to the left until the spring is com-
pressed 0.5 m and released from rest.
(a) If you assume that the disk rolls, what is its angular
acceleration at the instant it is released?
(b) What is the minimum coefficient of static friction
for which the disk will not slip when it is released?
kR
494
Problem 18.39 The disk weighs 12 lb and its radius
is 6 in. It is stationary on the surface when the force
F=10 lb is applied.
(a) If the disk rolls on the surface, what is the accel-
eration of its center?
(b) What minimum coefficient of static friction is nec-
essary for the disk to roll instead of slipping when
the force is applied?
F
32.2 ft/s2=8.94 ft/s2.a=17.9 ft/s2.
Problem 18.40 A 42-lb sphere with radius R=4in
is placed on a horizontal surface with initial angular
velocity ω0=40 rad/s. The coefficient of kinetic fric-
tion between the sphere and the surface is µk=0.06.
What maximum velocity will the center of the sphere
attain, and how long does it take to reach that velocity?
0
ω
496
Problem 18.41 A soccer player kicks the ball to a
teammate 8 m away. The ball leaves the player’s foot
moving parallel to the ground at 6 m/s with no angu-
lar velocity. The coefficient of kinetic friction between
the ball and the grass is µk=0.32. How long does it
take the ball to reach his teammate? The radius of the
ball is 112 mm and its mass is 0.4 kg. Estimate the ball’s
moment of inertia by using the equation for a thin spher-
ical shell: I=2
3mR2.
(b) Rolling — Steady motion
Problem 18.42 The 100-kg cylindrical disk is at rest
when the force Fis applied to a cord wrapped around
it. The static and kinetic coefficients of friction between
the disk and the surface equal 0.2. Determine the angular
acceleration of the disk if (a) F=500 N and (b) F=
1000 N.
Strategy: First solve the problem by assuming that the
disk does not slip, but rolls on the surface. Determine the
friction force, and find out whether it exceeds the product
of the coefficient of friction and the normal force. If it
does, you must rework the problem assuming that the
disk slips.
F
300 mm
From Newton’s second law: F−f=max, where axis the accelera-
(b) For F=1000 N the acceleration is
498
Problem 18.43 The ring gear is fixed. The mass and
moment of inertia of the sun gear are mS=320 kg and
IS=40 kg-m2. The mass and moment of inertia of each
planet gear are mP=38 kg and IP=0.60 kg-m2.Ifa
couple M=200 N-m is applied to the sun gear, what
is the latter’s angular acceleration? 0.50 m
M
0.86 m
Ring gear
0.18 m.
Planet Gears: Mc:Gr −Fr =IPαP
Ft:F+G=mPact
From kinematics act =−rαP
2αPrP=−RαS
We have 5 eqns in 5 unknowns. Solving, αS=3.95 rad/s2(counter-
clockwise)
GC
r
3 small disks
O
F
F
IP
Ms
Is
et
Problem 18.44 In Problem 18.43, what is the mag-
nitude of the tangential force exerted on the sun gear
by each planet gear at their point of contact when the
200 N-m couple is applied to the sun gear?
Problem 18.45 The 18-kg ladder is released from rest
in the position shown. Model it as a slender bar and
neglect friction. At the instant of release, determine
(a) the angular acceleration of the ladder and (b) the
normal force exerted on the ladder by the floor. (See
Active Example 18.3.)
4 m
30°
500
Problem 18.46 The 18-kg ladder is released from rest
in the position shown. Model it as a slender bar and
neglect friction. Determine its angular acceleration at the
instant of release.
20⬚
30⬚
4 m
Problem 18.47 The 4-kg slender bar is released from
rest in the position shown. Determine its angular
acceleration at that instant if (a) the surface is rough
and the bar does not slip, and (b) the surface is smooth.
1 m
502
Problem 18.48 The masses of the bar and disk are
14 kg and 9 kg, respectively. The system is released
from rest with the bar horizontal. Determine the bar’s
angular acceleration at that instant if
(a) the bar and disk are welded together at A,
(b) the bar and disk are connected by a smooth pin
at A.
Strategy: In part (b), draw individual free-body
diagrams of the bar and disk.
O
0.3 m
1.2 m
A
Solution:
Problem 18.49 The 5-lb horizontal bar is connected
to the 10-lb disk by a smooth pin at A. The system is
released from rest in the position shown. What are the
angular accelerations of the bar and disk at that instant?
A
3 ft 1 ft
O
Solution: Given
Ay
504
Problem 18.50 The 0.1-kg slender bar and 0.2-kg
cylindrical disk are released from rest with the bar
horizontal. The disk rolls on the curved surface. What is
the bar’s angular acceleration at the instant it is released?
40 mm
0.1 kg, Wb=mbg=0.981 N, md=0.2kg, Wd=mdg=1.962 N,
Problem 18.51 The mass of the suspended object Ais
8 kg. The mass of the pulley is 5 kg, and its moment of
inertia is 0.036 kg-m2. If the force T=70 N, what is
the magnitude of the acceleration of A?
120 mm
T
A
Solution: Given
The dynamic equations
FyB :T2+T−mBg−By=mBaBy
MB:−T2R+TR =IBαB
Kinematic constraints
aBy =aAy ,a
By =RαB
Solving we find aAy =0.805 m/s2
We also have
aBy =0.805 m/s2,α
B=6.70 rad/s,T
2=68.0 N,B
y=84.9 N
mAg
T
By
T2
506
c
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Problem 18.52 The suspended object Aweighs 20 lb.
The pulleys are identical, each weighing 10 lb and hav-
ing moment of inertia 0.022 slug-ft2. If the force T=
15 lb, what is the magnitude of the acceleration of A?
4 in
4 in
T
A
Solution: Given
The FBDs
Wdisk
T4
T2
Problem 18.53 The 2-kg slender bar and 5-kg block
are released from rest in the position shown. If fric-
tion is negligible, what is the block’s acceleration at that
instant? (See Example 18.5.) 1 m
508
