C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
334
89. In a demonstration known as the ballistics cart, a ball
is projected vertically upward from a cart moving with
constant velocity along the horizontal direction. The ball
lands in the catch
Problems
80. A thin rod of mass 0.630 kg and length 1.24 m is at rest,
hanging vertically from a strong xed hinge at its top
end. Suddenly a horizontal impulsive force (14.7 N is
i)
applied to it. (a) Suppose the force acts at the bottom
end of the rod. F
Problems
68. The speed of a moving bullet can be determined by allowing the bullet to pass through two rotating paper disks
mounted a distance d apart on the same axle (Fig. P10.68).
From the angular displacement of the two bullet holes
in the disks and t
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
73. As a result of friction, the angular speed of a wheel
changes with time according to
d
0e t
dt
of the disk, (b) the magnitude of the acceleration of the
center of mass is 2g/3, and (c) th
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
61. A long uniform rod of length L and mass M is pivoted
about a horizontal, frictionless pin through one end. The
rod is released from rest in a vertical position, as shown in
Figure P10.61.
Problems
329
friction required to maintain pure rolling motion for the
disk?
54. A uniform solid disk and a uniform hoop are placed side
by side at the top of an incline of height h. If they are released from rest and roll without slipping, which object
r
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
Pivot
M
R
g
R
M = 3.00 kg
R = 10.0 cm
m1 = 15.0 kg
m2 = 10.0 kg
m1
3.00 m
m2
Figure P10.46
47. This problem describes one experimental method for determining the moment of inertia of an irregu
Problems
43. In Figure P10.43 the sliding block has a mass of 0.850 kg,
the counterweight has a mass of 0.420 kg, and the pulley
is a hollow cylinder with a mass of 0.350 kg, an inner
radius of 0.020 0 m, and an outer radius of 0.030 0 m.
The coefcient of
326
31.
C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
Find the net torque on the wheel in Figure P10.31
about the axle through O if a 10.0 cm and b 25.0 cm.
10.0 N
30.0
a
O
12.0 N
37. A block of mass m1 2.00 kg and a block of mass
m 2 6.00 kg
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
Section 10.4 Rotational Kinetic Energy
20. Rigid rods of negligible mass lying along the y axis connect
three particles (Fig. P10.20). If the system rotates about
the x axis with an angular sp
Problems
25. A uniform thin solid door has height 2.20 m, width
0.870 m, and mass 23.0 kg. Find its moment of inertia for
rotation on its hinges. Is any piece of data unnecessary?
26. Attention! About face! Compute an order-of-magnitude estimate for the m
Problems
Cambridge, England, is at longitude 0, and Saskatoon,
Saskatchewan, is at longitude 107 west. How much time
elapses after the Pleiades set in Cambridge until these stars
fall below the western horizon in Saskatoon?
10. A merry-go-round is station
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
an incline (Fig. Q10.24). They are all released from rest at
the same elevation and roll without slipping. Which object
reaches the bottom rst? Which reaches it last? Try this at
home and note
Questions
321
The rate at which work is done by an external force in rotating a rigid object about
a xed axis, or the power delivered, is
(10.23)
If work is done on a rigid object and the only result of the work is rotation about a
xed axis, the net wor
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
The instantaneous angular speed of a particle moving in a circular path or of a
rigid object rotating about a xed axis is
d
dt
(10.3)
The instantaneous angular acceleration of a particle movin
Summary
319
Quick Quiz 10.12 A ball rolls without slipping down incline A, starting
from rest. At the same time, a box starts from rest and slides down incline B, which is
identical to incline A except that it is frictionless. Which arrives at the bottom
C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
318
P
P
v CM
v CM
CM
P
v=0
CM
v = R
v CM
(a) Pure translation
v = R
P
(b) Pure rotation
P
v = v CM + R = 2v CM
v = v CM
CM
P
v=0
(c) Combination of translation and rotation
Figure 10.29 The motion
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
Table 10.3
Useful Equations in Rotational and Linear Motion
Rotational Motion About a Fixed Axis
Linear Motion
Angular speed d/dt
Angular acceleration d/dt
Net torque
f i t
If
constant
f i i
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
potential energy. Because Ki 0 (the system is initially at
rest), we have
Solving for vf , we nd
Kf Uf Ki Ui
(1m 1vf 2
2
1
m v2
2 2 f
1
If 2)
2
1
2
1
m v2
2 2 f
m
1
1
2
m2
)gh
[m 2(mm m(I/R
S EC TI O N 10.9 Rolling Motion of a Rigid Object
R
317
s
Figure 10.27 For pure rolling motion, as
the cylinder rotates through an angle , its center
moves a linear distance s R.
s = R
The magnitude of the linear acceleration of the center of mass for pur
S ECTI O N 10.8 Work, Power, and Energy in Rotational Motion
Ei = U = MgL/2
O
1 2
I
2
0 1(1 ML2)2 0 1 MgL
2 3
2
L/2
O
1
Ef = KR = I 2
2
Figure 10.24 (Example 10.14) A uniform rigid rod pivoted at
O rotates in a vertical plane under the action of the
grav
C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
312
Because a/R, this expression can be simplied to
(m 1 m 2)g (m 1 m 2)a 2I
(7)
a
a
R2
(m1 m2)g
m1 m2 2(I/R 2)
T1 m 1g m 1a m 1(g a)
2m 1g
(m 1 m 2)g
m 1 m 2 2(I/R 2)
m
m 2 (I/R 2)
2
1 m 2 2(I/R
S ECTI O N 10.7 Relationship Between Torque and Angular Acceleration
Substituting Equation (4) into Equation (2) and solving for
a and , we nd that
(5)
a
We can show this mathematically by taking the limit
I : , so that Equation (5) becomes
g
1 (I/mR 2 )
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
Conceptual Example 10.11
Falling Smokestacks and Tumbling Blocks
When a tall smokestack falls over, it often breaks somewhere
along its length before it hits the ground, as shown in Figure
10.
S ECTI O N 10.8 Work, Power, and Energy in Rotational Motion
313
Because the magnitude of the torque due to F about O is dened as rF sin by
Equation 10.19, we can write the work done for the innitesimal rotation as
dW d
(10.22)
The rate at which work is b
S ECTI O N 10.7 Relationship Between Torque and Angular Acceleration
309
Quick Quiz 10.10
You turn off your electric drill and nd that the time interval for the rotating bit to come to rest due to frictional torque in the drill is t. You
replace the bit w
C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
306
F sin
F
r
O
r
F cos
Line of
action
d
Figure 10.13 The force F has a
greater rotating tendency about O
as F increases and as the moment
arm d increases. The component
F sin tends to rotate th
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308
Consider a particle of mass m rotating in a circle of radius r under the inuence of
a tangential force Ft and a radial force Fr , as shown in Figure 10.16. The tangential
force provides a tang
S ECTI O N 10.7 Relationship Between Torque and Angular Acceleration
307
Quick Quiz 10.8 If you are trying to loosen a stubborn screw from a piece
of wood with a screwdriver and fail, should you nd a screwdriver for which the handle
is (a) longer or (b) f
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C HAPTE R 10 Rotation of a Rigid Object About a Fixed Axis
Answer Note that the result for the moment of inertia of a
cylinder does not depend on L, the length of the cylinder.
In other words, it applies equally well to a long cylinder and
a at disk h