Physics for Scientists and Engineers
Storyline
Tenth Edition
Raymond A. Serway & John W. Jewett, Jr., Physics for Scientists and Engineers, Tenth Edition. © 2021 Cengage. All Rights
Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 10
Rotation of a Rigid Object About a
Fixed Axis
Raymond A. Serway & John W. Jewett, Jr., Physics for Scientists and Engineers, Tenth Edition. © 2021 Cengage. All Rights
Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Angular Position (1 of 2)
s
q=
r
s = rq
2p r
C = 2p r Þ 360° corresponds to
= 2p rad
r
360°
p
1 rad =
» 57.3°
q ( rad ) =
q ( deg )
2p
180°
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accessible website, in whole or in part.
Angular Position (2 of 2)
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accessible website, in whole or in part.
Angular Speed
Dq º q f - q i
wavg
Dq
º
Dt
Dq dq
w º lim
=
Dt ® 0 Dt
dt
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accessible website, in whole or in part.
Quick Quiz 10.1 Part I (1 of 2)
A rigid object rotates in a counterclockwise sense around a fixed axis. Each of
the following pairs of quantities represents an initial angular position and a
final angular position of the rigid object.
Which of the sets can only occur if the rigid object rotates through more than
180°?
(a) 3 rad, 6 rad
(b) –1 rad, 1 rad
(c) 1 rad, 5 rad
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accessible website, in whole or in part.
Quick Quiz 10.1 Part I (2 of 2)
A rigid object rotates in a counterclockwise sense around a fixed axis. Each of
the following pairs of quantities represents an initial angular position and a
final angular position of the rigid object.
Which of the sets can only occur if the rigid object rotates through more than
180°?
(a) 3 rad, 6 rad
(b) –1 rad, 1 rad
(c) 1 rad, 5 rad
© 2021 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly
accessible website, in whole or in part.
Quick Quiz 10.1 Part II (1 of 2)
A rigid object rotates in a counterclockwise sense around a fixed axis. Each of
the following pairs of quantities represents an initial angular position and a
final angular position of the rigid object.
Suppose the change in angular position for each of these pairs of values
occurs in 1 s. Which choice represents the lowest average angular speed?
(a) 3 rad, 6 rad
(b) –1 rad, 1 rad
(c) 1 rad, 5 rad
© 2021 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly
accessible website, in whole or in part.
Quick Quiz 10.1 Part II (2 of 2)
A rigid object rotates in a counterclockwise sense around a fixed axis. Each of
the following pairs of quantities represents an initial angular position and a
final angular position of the rigid object.
Suppose the change in angular position for each of these pairs of values
occurs in 1 s. Which choice represents the lowest average angular speed?
(a) 3 rad, 6 rad
(b) –1 rad, 1 rad
(c) 1 rad, 5 rad
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accessible website, in whole or in part.
Average and Instantaneous Angular Acceleration
a avg
Dw w f - wi
º
=
Dt
t f - ti
Dw d w
a º lim
=
Dt ® 0 Dt
dt
q ®x w ®v a ®a
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accessible website, in whole or in part.
Directions of Angular Vectors
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accessible website, in whole or in part.
Analysis Model: Rigid Object Under Constant Angular
Acceleration (1 of 3)
t
dw
a=
Þ d w = a dt Þ Dw = ò a dt
0
dt
w f = wi + a t
( for constant a )
t
dq
w=
Þ dq = w dt Þ Dq = ò (wi + a t ) dt
0
dt
1 2
q f =qi +wi t + a t
2
( for constant a )
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accessible website, in whole or in part.
Analysis Model: Rigid Object Under Constant Angular
Acceleration (2 of 3)
1 2
w f = wi + a t and q f =qi +wi t + a t
2
w f - wi
æ w f - wi
t=
® q f =qi +wi ç
a
è a
ö 1 æ w f - wi ö
÷+ aç
÷
ø 2 è a ø
2
2a (q f - q i ) = 2wi (w f - wi ) + (w f - wi ) Þ
2
2a (q f - q i ) = 2wiw f - 2wi 2 + w f 2 + wi 2 - 2wiw f
w f 2 =wi 2 +2a (q f - qi )
( for constant a )
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accessible website, in whole or in part.
Analysis Model: Rigid Object Under Constant Angular
Acceleration (3 of 3)
1 2
w f = wi + a t and q f =qi +wi t + a t
2
w f - wi
1 æ w f - wi ö 2
a=
Þ q f =qi +wi t + ç
÷t
t
2è
t
ø
q f =qi +wi t +
w f t wi t
1
q f =qi + (wi + w f ) t
2
2
-
2
Þ
( for constant a )
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accessible website, in whole or in part.
Kinematic Expressions
Table 10.1 Kinematic Equations for Rotational and Translational Motion
Rigid Object Under Constant Angular Acceleration
Particle Under Constant Acceleration
w f = wi + a t
(10.6 )
v f = vi + at
( 2.13)
1 2
q f = qi + wi t + a t
2
(10.7 )
( 2.16 )
w 2f = wi2 + 2a (q f - q i )
1 2
x f = xi + vi t + at
2
(10.8)
v 2f = vi2 + 2a ( x f - xi )
( 2.17 )
(10.9 )
1
x f = xi + ( vi + v f ) t
2
( 2.15)
1
q f = qi + (wi + w f ) t
2
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accessible website, in whole or in part.
Quick Quiz 10.2 (1 of 2)
A rigid object rotates in a counterclockwise sense around a fixed axis. If the
object starts from rest at the initial angular position, moves counterclockwise
with constant angular acceleration, and arrives at the final angular position
with the same angular speed in all three cases, for which choice is the angular
acceleration the highest?
(a) 3 rad, 6 rad
(b) –1 rad, 1 rad
(c) 1 rad, 5 rad
© 2021 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly
accessible website, in whole or in part.
Quick Quiz 10.2 (2 of 2)
A rigid object rotates in a counterclockwise sense around a fixed axis. If the
object starts from rest at the initial angular position, moves counterclockwise
with constant angular acceleration, and arrives at the final angular position
with the same angular speed in all three cases, for which choice is the angular
acceleration the highest?
(a) 3 rad, 6 rad
(b) –1 rad, 1 rad
(c) 1 rad, 5 rad
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accessible website, in whole or in part.
Analysis Model: Rigid Object Under Constant Angular
Acceleration
w f = wi + a t
1 2
q f =qi +wi t + a t
2
2
2
w f =wi +2a (q f - qi )
1
q f =qi + (wi + w f ) t
2
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accessible website, in whole or in part.
Example 10.1: Rotating Wheel (1 of 5)
A wheel rotates with a constant angular acceleration
of 3.50 rad/s2.
(A) If the angular speed of the wheel is 2.00 rad/s at ti
= 0, through what angular displacement does the
wheel rotate in 2.00 s?
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accessible website, in whole or in part.
Example 10.1: Rotating Wheel (2 of 5)
1 2
Dq = q f - qi = wi t + a t
2
1
2
2
Dq = ( 2.00 rad/s )( 2.00 s ) + ( 3.50 rad/s ) ( 2.00 s )
2
= 11.0 rad = (11.0 rad )(180°/p rad ) = 630°
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accessible website, in whole or in part.
Example 10.1: Rotating Wheel (3 of 5)
(B) Through how many revolutions has the wheel turned during this time
interval?
æ 1 rev ö
Dq = 630° ç
÷ = 1.75 rev
è 360° ø
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accessible website, in whole or in part.
Example 10.1: Rotating Wheel (4 of 5)
(C) What is the angular speed of the wheel at t = 2.00 s?
w f = wi + a t
= 2.00 rad/s+ ( 3.50 rad/s 2 ) ( 2.00 s ) = 9.00 rad/s
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accessible website, in whole or in part.
Example 10.1: Rotating Wheel (5 of 5)
Suppose a particle moves along a straight line with a constant acceleration of
3.50 m/s2. If the velocity of the particle is 2.00 m/s at ti = 0, through what
displacement does the particle move in 2.00 s? What is the velocity of the
particle at t = 2.00 s?
1 2
Dx = x f - xi = vi t + at
2
1
2
2
= ( 2.00 m/s )( 2.00 s ) + ( 3.50 m/s ) ( 2.00 s ) = 11.0 m
2
v f = vi + at = 2.00 m/s + ( 3.50 m/s
2
) ( 2.00 s )
2
= 9.00 m/s
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accessible website, in whole or in part.
Angular and Translational Quantities (1 of 2)
s = rq
ds
dq
v=
=r
dt
dt
v = rw
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accessible website, in whole or in part.
Angular and Translational Quantities (2 of 2)
dv d
dw
at =
= ( rw ) = r
dt dt
dt
at = ra
v2
ac =
= rw 2
r
  
a = at + a r
a = at + ar = r a + r w = r a + w
2
2
2
2
2
4
2
4
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accessible website, in whole or in part.
Quick Quiz 10.3 Part I (1 of 2)
Ethan and Rebecca are riding on a merry-go-round. Ethan rides on a horse at
the outer rim of the circular platform, twice as far from the center of the circular
platform as Rebecca, who rides on an inner horse.
When the merry-go-round is rotating at a constant angular speed, what is
Ethan’s angular speed?
(a) twice Rebecca’s
(b) the same as Rebecca’s
(c) half of Rebecca’s
(d) impossible to determine
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accessible website, in whole or in part.
Quick Quiz 10.3 Part I (2 of 2)
Ethan and Rebecca are riding on a merry-go-round. Ethan rides on a horse at
the outer rim of the circular platform, twice as far from the center of the circular
platform as Rebecca, who rides on an inner horse.
When the merry-go-round is rotating at a constant angular speed, what is
Ethan’s angular speed?
(a) twice Rebecca’s
(b) the same as Rebecca’s
(c) half of Rebecca’s
(d) impossible to determine
© 2021 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly
accessible website, in whole or in part.
Quick Quiz 10.3 Part II (1 of 2)
Ethan and Rebecca are riding on a merry-go-round. Ethan rides on a horse at
the outer rim of the circular platform, twice as far from the center of the circular
platform as Rebecca, who rides on an inner horse.
When the merry-go-round is rotating at a constant angular speed, describe
Ethan’s tangential speed.
(a) twice Rebecca’s
(b) the same as Rebecca’s
(c) half of Rebecca’s
(d) impossible to determine
© 2021 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly
accessible website, in whole or in part.
Quick Quiz 10.3 Part II (2 of 2)
Ethan and Rebecca are riding on a merry-go-round. Ethan rides on a horse at
the outer rim of the circular platform, twice as far from the center of the circular
platform as Rebecca, who rides on an inner horse.
When the merry-go-round is rotating at a constant angular speed, describe
Ethan’s tangential speed.
(a) twice Rebecca’s
(b) the same as Rebecca’s
(c) half of Rebecca’s
(d) impossible to determine
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accessible website, in whole or in part.
Example 10.2 CD Player (1 of 2)
Despite the availability of music in digital form, the
compact disc, or CD, remains a popular format for
music and data. On a CD, audio information is
stored digitally in a series of pits and flat areas on
the surface of the disc. The alternations between
pits and flat areas on the surface represent binary
ones and zeros to be read by the CD player and
converted back to sound waves. The pits and flat
areas are detected by a system consisting of a
laser and lenses. The length of a string of ones and
zeros representing one piece of information is the
same everywhere on the disc, whether the
information is near the center of the disc or near its
outer edge.
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accessible website, in whole or in part.
Example 10.2 CD Player (2 of 2)
So that this length of ones and zeros always passes by
the laser–lens system in the same time interval, the
tangential speed of the disc surface at the location of
the lens must be constant. According to v=rw, the
angular speed must therefore vary as the laser–lens
system moves radially along the disc. In a typical CD
player, the constant speed of the surface at the point of
the laser–lens system is 1.3 m/s.
(A) Find the angular speed of the disc in revolutions per
minute when information is being read from the
innermost first track (r = 23 mm) and the outermost final
track (r = 58 mm).
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accessible website, in whole or in part.
Example 10.2: CD Player (1 of 3)
v
1.3 m/s
wi = =
= 57 rad/s
-2
ri 2.3 ´ 10 m
æ 1 rev öæ 60 s ö
2
= ( 57 rad/s ) ç
=
5.4
´
10
rev/min
֍
÷
è 2p rad øè 1 min ø
v
1.3 m/s
wf = =
rf 5.8 ´ 10-2 m
= 22 rad/s = 2.1´ 10 rev/min
2
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accessible website, in whole or in part.
Example 10.2: CD Player (2 of 3)
(B) The maximum playing time of a standard music disc is 74 min and 33 s.
How many revolutions does the disc make during that time?
t = ( 74 min )( 60 s/min ) + 33 s = 4 473 s
1
Dq = q f - q i = (wi + w f ) t
2
1
5
= ( 57 rad/s + 22 rad/s )( 4 473 s ) = 1.8 ´ 10 rad
2
æ 1 rev ö
4
Dq = (1.8 ´ 10 rad ) ç
=
2.8
´
10
rev
÷
è 2p rad ø
5
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accessible website, in whole or in part.
Example 10.2: CD Player (3 of 3)
(C) What is the angular acceleration of the compact disc over the 4 473-s time
interval?
a=
w f - wi
t
22 rad/s - 57 rad/s
-3
2
=
= -7.6 ´ 10 rad/s
4 473 s
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accessible website, in whole or in part.
Torque (1 of 2)
Torque (t ): vector quantity measuring
changes in rotational motion of object about
some axis
t º rF sin q = Fd
d = r sin f
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accessible website, in whole or in part.
Torque (2 of 2)
åt = t
1
+t 2
= F1d1 - F2 d 2
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accessible website, in whole or in part.
Quick Quiz 10.4 (1 of 2)
You are trying to loosen a stubborn screw from a piece of wood with a
screwdriver and fail,. You should find a screwdriver for which the handle is
(a) longer.
(b) wider.
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accessible website, in whole or in part.
Quick Quiz 10.4 (2 of 2)
You are trying to loosen a stubborn screw from a piece of wood with a
screwdriver and fail. You should find a screwdriver for which the handle is
(a) longer.
(b) wider.
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accessible website, in whole or in part.
Example 10.3: The Net Torque on a Cylinder (1 of 3)
A one-piece cylinder is shaped as shown in the figure,
with a core section protruding from the larger drum.
The cylinder is free to rotate about the central z axis
shown in the drawing. A rope wrapped
 around the drum,
which has radius R1 , exerts a force T1 to the right on the
cylinder. A rope wrapped
 around the core, which has
radius R2 , exerts a force T2 downward
on the cylinder.
(A) What is the net torque acting on the
cylinder about the rotation axis (which is
the z axis in the figure)?
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accessible website, in whole or in part.
Example 10.3: The Net Torque on a Cylinder (2 of 3)
åt = t
1
+ t 2 = R2T2 - R1T1
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accessible website, in whole or in part.
Example 10.3: The Net Torque on a Cylinder (3 of 3)
(B) Suppose T1 = 5.0 N, R1 = 1.0 m, T2 = 15 N, and R2 = 0.50 m. What is the
net torque about the rotation axis, and which way does the cylinder rotate
starting from rest?
åt = ( 0.50 m )(15 N ) - (1.0 m )( 5.0 N ) =
2.5 N × m
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accessible website, in whole or in part.
Analysis Model: Rigid Object Under a Net Torque (1 of 3)
åF
= mat
t
åt = å F r = ( ma ) r
t
t
åt = ( mra ) r = ( mr )a
2
åt = I a
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accessible website, in whole or in part.
Analysis Model: Rigid Object Under a Net Torque (2 of 3)
Fi = mi ai
t i = ri Fi = ri mi ai
t i = mi ri a
2
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accessible website, in whole or in part.
Analysis Model: Rigid Object Under a Net Torque (3 of 3)
åt
æ
2ö
= åt i = å mi ri a = ç å mi ri ÷ a
i
i
è i
ø
2
ext
åt
ext
= Ia
where I = å mi ri
2
i
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accessible website, in whole or in part.
Moment of Inertia (1 of 2)
I = å mi ri
2
i
åt
ext
= Ia


å Fext = MaCM
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accessible website, in whole or in part.
Moment of Inertia (2 of 2)
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accessible website, in whole or in part.
Quick Quiz 10.5 (1 of 2)
You turn off your electric drill and find that the time interval for the rotating bit
to come to rest due to frictional torque in the drill is Dt. You replace the bit with
a larger one that results in a doubling of the moment of inertia of the drill’s
entire rotating mechanism. When this larger bit is rotated at the same angular
speed as the first and the drill is turned off, the frictional torque remains the
same as that for the previous situation. What is the time interval for this
second bit to come to rest?
(a) 4Dt
(d) 0.5Dt
(b) 2Dt
(e) 0.25Dt
(c) Dt
(f) impossible to determine
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accessible website, in whole or in part.
Quick Quiz 10.5 (2 of 2)
You turn off your electric drill and find that the time interval for the rotating bit
to come to rest due to frictional torque in the drill is Dt. You replace the bit with
a larger one that results in a doubling of the moment of inertia of the drill’s
entire rotating mechanism. When this larger bit is rotated at the same angular
speed as the first and the drill is turned off, the frictional torque remains the
same as that for the previous situation. What is the time interval for this
second bit to come to rest?
(a) 4Dt
(b) 2Dt
(c) Dt
(d) 0.5Dt
(e) 0.25Dt
(f) impossible to determine
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accessible website, in whole or in part.
Analysis Model: Rigid Object Under a Net Torque
åt
ext
= Ia
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accessible website, in whole or in part.
Example 10.4: Rotating Rod (1 of 2)
A uniform rod of length L and mass M is attached at one end to a frictionless
pivot and is free to rotate about the pivot in the vertical plane as in the figure.
The rod is released from rest in the horizontal position.
What are the initial angular acceleration of the rod and the initial translational
acceleration of its right end?
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accessible website, in whole or in part.
Example 10.4: Rotating Rod (2 of 2)
åt
t
å
a=
I
ext
=
ext
Mg ( L /2 )
1
3
ML2
æLö
= Mg ç ÷
è2ø
3g
=
2L
3
at = La = g
2
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accessible website, in whole or in part.
Conceptual Example 10.5: 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 the figure. Why?
at = ra
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accessible website, in whole or in part.
Example 10.6: Angular Acceleration of a Wheel (1 of 3)
A wheel of radius R, mass M, and moment of inertia
I is mounted on a frictionless, horizontal axle as in
the figure. A light cord wrapped around the wheel
supports an object of mass m. When the wheel is
released, the object accelerates downward, the cord
unwraps off the wheel, and the wheel rotates with an
angular acceleration. Find expressions for the
angular acceleration of the wheel, the translational
acceleration of the object, and the tension in the
cord.
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accessible website, in whole or in part.
Example 10.6: Angular Acceleration of a Wheel (2 of 3)
åt ext = Ia
å Fy = mg - T = ma
a = Ra
T=
mg
1 + ( mR /I )
2
a=
t
å
®a =
ext
I
=
TR
I
mg - T
®a=
m
TR 2 mg - T
=
=
I
m
a=
g
1 + ( I /mR 2 )
a
g
=
R R + ( I /mR )
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accessible website, in whole or in part.
Example 10.6: Angular Acceleration of a Wheel (3 of 3)
What if the wheel were to become very massive so that I becomes very large?
What happens to the acceleration a of the object and the tension T ?
a = lim
I ®¥
T = lim
I ®¥
g
1 + ( I /mR 2 )
mg
1 + ( mR 2 /I )
=0
= mg
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accessible website, in whole or in part.
Calculation of Moments of Inertia (1 of 3)
I = å ri 2 Dmi ( system of discrete particles )
i
Continuous rigid object: I = lim
Dmi ® 0
m
r=
V
år
i
i
2
Dmi = ò r dm
2
Þ dm = r dV
I = ò r r dV
2
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accessible website, in whole or in part.
Calculation of Moments of Inertia (2 of 3)
r = m /V
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accessible website, in whole or in part.
Calculation of Moments of Inertia (3 of 3)
m
s = = r t ( mass per unit area )
A
m
l = = r A ( mass per unit length )
L
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accessible website, in whole or in part.
Example 10.7: Uniform Rigid Rod
Calculate the moment of inertia of a uniform thin rod of length L and mass M
about an axis perpendicular to the rod (the y axis) and passing through its
center of mass.
M
dm = l dx =
dx
L
M
I y = ò r dm = ò x
dx
-L/2
L
M L/2 2
=
x dx
ò
L
/
2
L
3 L/2
M éx ù
1
2
=
=
ML
ê ú
L ë 3 û - L /2 12
2
L/2
2
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accessible website, in whole or in part.
Example 10.8: Uniform Solid Cylinder (1 of 2)
A uniform solid cylinder has a radius R, mass M, and length L. Calculate its
moment of inertia about its central axis (the z axis in in the figure).
dV = LdA = L ( 2p r ) dr
dm = p dV = p L ( 2p r ) dr
I z = ò r 2 dm = ò r 2 éë r L ( 2p r ) dr ùû
R
1
= 2pr L ò r 3 dr = pr LR 4
0
2
r=
M
M
=
V p R2 L
1 æ M ö 4 1
2
Iz = p ç
LR
=
MR
÷
2 è p R2 L ø
2
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accessible website, in whole or in part.
Example 10.8: Uniform Solid Cylinder (2 of 2)
What if the length of the cylinder in the figure is increased to 2L, while the
mass M and radius R are held fixed (The density becomes half as large)?
How does that change the moment of
inertia of the cylinder?
1
2
I z = MR
2
Icylinder is not affected
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accessible website, in whole or in part.
Parallel-Axis Theorem (1 of 3)
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accessible website, in whole or in part.
Parallel-Axis Theorem (2 of 3)
r=
( x¢ )
2
+ ( y¢)
2
2
2
2
I = ò ( r ¢ ) dm = ò é( x¢ ) + ( y ¢ ) ù dm
ë
û
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accessible website, in whole or in part.
Parallel-Axis Theorem (3 of 3)
¢
¢ z¢ = z = 0
x¢ = x + xCM
y ¢ = y + yCM
2
2
¢ ) + ( y + yCM
¢ ) ù dm
I = ò é( x + xCM
ë
û
¢ ò x dm
= ò ( x 2 + y 2 ) dm + 2 xCM
¢ ò y dm + ( xCM
¢ 2 + yCM
¢ 2 ) ò dm
+ 2 yCM
ò x dm = ò y dm = 0
2
2
2
¢
¢
dm
=
M
and
D
=
x
+
y
CM
CM
ò
I = I CM + MD 2
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accessible website, in whole or in part.
Example 10.9: Applying the Parallel-Axis Theorem
Consider once again the uniform rigid rod of mass M and length L shown in
the figure. Find the moment of inertia of the rod about an axis perpendicular to
the rod through one end (the y axis in the figure).
I = I CM + MD
2
1
æLö
2
= ML + M ç ÷
12
è2ø
1
2
= ML
3
2
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accessible website, in whole or in part.
Rotational Kinetic Energy
1
K i = mi vi 2
2
1
1
2
K R = å K i =å mi vi = å mi ri 2w 2
2 i
i
i 2
1æ
2ö 2
K R = ç å mi ri ÷ w
2è i
ø
1 2
K R = Iw
2
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accessible website, in whole or in part.
Quick Quiz 10.6 (1 of 2)
A section of hollow pipe and a solid cylinder have the same radius, mass, and
length. They both rotate about their long central axes with the same angular
speed. Which object has the higher rotational kinetic energy?
(a) The hollow pipe does.
(b) The solid cylinder does.
(c) They have the same rotational kinetic energy.
(d) It is impossible to determine.
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accessible website, in whole or in part.
Quick Quiz 10.6 (2 of 2)
A section of hollow pipe and a solid cylinder have the same radius, mass, and
length. They both rotate about their long central axes with the same angular
speed. Which object has the higher rotational kinetic energy?
(a) The hollow pipe does.
(b) The solid cylinder does.
(c) They have the same rotational kinetic energy.
(d) It is impossible to determine.
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accessible website, in whole or in part.
Example 10.10: An Unusual Baton (1 of 5)
Four tiny spheres are fastened to the ends of two
rods of negligible mass lying in the xy plane to form
an unusual baton (see figure). We shall assume
the radii of the spheres are small compared with
the dimensions of the rods.
(A) If the system rotates about the y axis with an
angular speed w, find the moment of inertia and the
rotational kinetic energy of the system about this
axis.
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accessible website, in whole or in part.
Example 10.10: An Unusual Baton (2 of 5)
I y = å mi ri = Ma + Ma = 2Ma
2
2
2
2
i
1
2
K R = I yw
2
1
2
2
2 2
= ( 2 Ma ) w = Ma w
2
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accessible website, in whole or in part.
Example 10.10: An Unusual Baton (3 of 5)
(B) Suppose the system rotates in the xy plane about an axis (the z axis)
through the center of the baton. Calculate the moment of inertia and rotational
kinetic energy about this axis.
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accessible website, in whole or in part.
Example 10.10: An Unusual Baton (4 of 5)
I z = å mi ri 2 = Ma 2 + Ma 2 + mb 2 + mb 2
i
= 2 Ma + 2mb
2
2
1
2
K R = I zw
2
1
2
2
2
= ( 2 Ma + 2mb ) w
2
2
2
2
= ( Ma + mb ) w
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accessible website, in whole or in part.
Example 10.10: An Unusual Baton (5 of 5)
What if the mass M is much larger than m? How do the answers to parts (A)
and (B) compare?
I z = 2Ma and K R = Ma w
2
2
2
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accessible website, in whole or in part.
Energy Considerations in Rotational Motion (1 of 2)
 
dW = F × d s = ( F sin f ) rdq
dW = t dq
dW
dq
=t
dt
dt
dW
P=
= tw
dt
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accessible website, in whole or in part.
Energy Considerations in Rotational Motion (2 of 2)
åt
åt
ext
ext
= Ia
dw
d w dq
dw
= Ia = I
=I
=I
w
dt
dq dt
dq
åt
W =ò
wf
wi
ext
dq = dW = I w d w
1
1
2
I w d w = I w f - I wi 2
2
2
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accessible website, in whole or in part.
Rotational and Translational Motion Expressions
Table 10.3 Useful Equations in Rotational and Translational Motion
Rotational Motion About a Fixed Axis
Translational Motion
Angular speed w = dq / dt
Translational speed v = dx / dt
Angular acceleration a = d w / dt
Net torque St ext = I a
Translational acceleration a = dv / dt
Net force SF = ma
If
If
ì
w f = wi + a t
ï
1
ï
a = constant í q = qi + wi t + a t 2
2
ï
ïw 2f = wi2 + 2a (q f - q i )
î
ì
v f = vi + a t
ï
1
ï
a = constant í x f = xi + vi t + a t 2
2
ï
ïv 2f = vi2 + 2a ( x f - xi )
î
xf
Work W = ò t dq
Work W = ò Fx dx
1
Rotational kinetic energy K R = I w 2
2
Power P = tw
kinetic energy K =
qf
qi
Angular momentum L = I w
Net torque St = dL / dt
xi
1 2
mv
2
Power P = Fv
Linear momentum p = mv
Net force SF = dp / dt
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accessible website, in whole or in part.
Example 10.11: Rotating Rod Revisited (1 of 4)
A uniform rod of length L and mass M is free to rotate on a frictionless pin
passing through one end. The rod is released from rest in the horizontal
position.
(A) What is its angular speed when the rod reaches its lowest position?
DK + DU = 0
1
æ1 2
ö æ
ö
ç I w - 0 ÷ + ç 0 - mgL ÷ = 0
2
è2
ø è
ø
w=
MgL
=
I
MgL
=
2
1
3 ML
3g
L
© 2021 Cengage. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly
accessible website, in whole or in part.
Example 10.11: Rotating Rod Revisited (2 of 4)
(B) Determine the tangential speed of the center of mass and the tangential
speed of the lowest point on the rod when it is in the vertical position.
vCM
L
1
= rw = w =
3gL
2
2
v = 2vCM =
3gL
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accessible website, in whole or in part.
Example 10.11: Rotating Rod Revisited (3 of 4)
What if we want to find the angular speed of the rod when the angle it makes
with the horizontal is 45.0°? Because this angle is half of 90.0°, for which we
solved the problem in the previous slide, is the angular speed at this
configuration half the previous answer, that is, 1
3 g L?
2
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accessible website, in whole or in part.
Example 10.11: Rotating Rod Revisited (4 of 4)
L
2L
cos 45° = h =
2
4
1 2
0 = I w - Mgh
2
æ 2L ö
1æ1
2ö 2
= ç ML ÷ w - Mg çç
÷÷
2è3
ø
è 4 ø
1
w = 1/ 4
2
3g
3g
» 0.841
L
L
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accessible website, in whole or in part.
Example 10.12: Energy and the Atwood Machine (1 of 2)
Two blocks having different masses m1 and m2 are
connected by a string passing over a pulley. The
pulley has a radius R and moment of inertia I about
its axis of rotation. The string does not slip on the
pulley, and the system is released from rest. Find the
translational speeds of the blocks after block 2
descends through a distance h and find the angular
speed of the pulley at this time.
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accessible website, in whole or in part.
Example 10.12: Energy and the Atwood Machine (2 of 2)
DK + DU = 0
éæ 1
1
1
ù
2
2
2ö
m
v
+
m
v
+
I
w
0
+ éë( m1 gh - m2 gh ) - 0 ùû = 0
2 f
f ÷
êç 2 1 f
ú
2
2
ø û
ëè
2
1
1
1 vf
2
2
m1v f + m2 v f + I 2 = m2 gh - m1 gh
2
2
2 R
1æ
I ö 2
m
+
m
+
v = ( m2 - m1 ) gh
2
ç 1
2 ÷ f
2è
R ø
é 2 ( m2 - m1 ) gh ù
vf = ê
2 ú
m
+
m
+
I
/
R
2
ë 1
û
1/ 2
1 é 2 ( m2 - m1 ) gh ù
wf =
= ê
ú
R
R ë m1 + m2 + I /R 2 û
vf
1/ 2
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accessible website, in whole or in part.
Rolling Motion of a Rigid Object (1 of 2)
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accessible website, in whole or in part.
Rolling Motion of a Rigid Object (2 of 2)
vCM
aCM
ds
dq
=
=R
= Rw
dt
dt
dvCM
dw
=
=R
= Ra
dt
dt
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accessible website, in whole or in part.
Kinetic Energy of a Rolling Object
1
2
K = I Pw
2
1
1
2
2 2
K = I CMw + MR w
2
2
1
1
2
K = I CMw + MvCM 2
2
2
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accessible website, in whole or in part.
Analysis of Rolling Object
2
1
1 æ I CM
æ vCM ö 1
ö
2
2
K = I CM ç
+
Mv
Þ
K
=
+
M
v
÷ 2 CM
ç R2
÷ CM
2
R
2
è
ø
è
ø
é 1 æ I CM
ù
ö
2
DK + DU = 0 Þ ê ç 2 + M ÷ vCM - 0 ú + ( 0 - Mgh ) = 0
ø
ë2 è R
û
1/ 2
vCM
é
ù
2 gh
ú
=ê
2
êë1 + ( I CM /MR ) úû
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accessible website, in whole or in part.
Quick Quiz 10.7 (1 of 2)
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 first?
(a) The ball arrives first.
(b) The box arrives first.
(c) Both arrive at the same time.
(d) It is impossible to determine.
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accessible website, in whole or in part.
Quick Quiz 10.7 (2 of 2)
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 first?
(a) The ball arrives first.
(b) The box arrives first.
(c) Both arrive at the same time.
(d) It is impossible to determine.
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accessible website, in whole or in part.
Example 10.13: Sphere Rolling Down and Incline (1 of 2)
Suppose the sphere shown in the figure is solid and uniform. Calculate the
translational speed of the center of mass at the bottom of the incline and the
magnitude of the translational acceleration of the center of mass.
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accessible website, in whole or in part.
Example 10.13: Sphere Rolling Down and Incline (2 of 2)
1/ 2
vCM
é
ù
2 gh
ú
=ê
2
2
2
êë1 + ( 5 MR /MR ) úû
1/ 2
æ 10 ö
= ç gh ÷
è7
ø
h = x sin q
vCM
2
10
= gx sin q
7
vCM 2 = 2aCM x
aCM
5
= g sin q
7
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accessible website, in whole or in part.
Example 10.14: Pulling on a Spool (1 of 3)
A cylindrically symmetric spool of mass m and radius R sits at rest on a horizontal
table with friction. With your hand on a light string wrapped around the axle of
radius r, you pull on the spool with a constant horizontal force of magnitude T to
the right. As a result, the spool rolls without slipping a distance L along the table
with no rolling friction.
(A) Find the final translational speed of the center of mass of the spool.
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accessible website, in whole or in part.
Example 10.14: Pulling on a Spool (2 of 3)
W = DK = DK trans + DK rot
 + L = L (1 + r /R )
r
 = rq = L
R
rö
æ
1
1
W = TL ç1 + ÷ = mvCM 2 + I w 2
2
2
è Rø
2
v
rö 1
1
æ
TL ç1 + ÷ = mvCM 2 + I CM2
2 R
è Rø 2
vCM =
2TL (1 + r /R )
m (1 + I /mR 2 )
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accessible website, in whole or in part.
Example 10.14: Pulling on a Spool (3 of 3)
(B) Find the value of the friction force f.
m ( vCM - 0 ) = (T - f ) Dt ® mvCM = (T - f ) Dt
Dt =
mvCM
L
vCM,avg
2L
= (T - f )
vCM
2L
=
vCM
mvCM
® f =T 2L
2
1 + r /R )
(
m é 2TL (1 + r /R ) ù
é 1 - mrR ù
ê
ú = T -T
f =T =T ê
2
2
2 L ê m (1 + I /mR ) ú
1 + mR 2 ûú
1
+
I
/
mR
ë
(
)
ë
û
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accessible website, in whole or in part.
Assessing to Learn (1 of 11)
The rotational inertia of the dumbbell (see figure) about axis A is twice the
rotational inertia about axis B. The unknown mass is:
1. 4/7 kg
3. 4 kg
5. 7 kg
2. 2 kg
4. 5 kg
6. 8 kg
7. 10 kg
8. None of the above
9. Cannot be determined
10. The rotational inertia cannot be different about different axes.
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accessible website, in whole or in part.
Assessing to Learn (2 of 11)
A disk, with radius 0.25 m and mass 4 kg, lies flat on a smooth horizontal
tabletop. A string wound about the disk is pulled with a force of 8 N. What is the
acceleration of the disk?
1. 0
2. 2 m/s²
3. 4 m/s²
4. 8 m/s²
5. 16 m/s²
6. None of the above.
7. Cannot be determined.
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accessible website, in whole or in part.
Assessing to Learn (3 of 11)
A disk, with radius 0.25 m and mass 4 kg, lies flat on a smooth horizontal
tabletop. A string wound about the disk is pulled with a force of 8 N. What is the
angular acceleration of the disk?
1. 0
2. 2 rad/s²
3. 4 rad/s²
4. 8 rad/s²
5. 16 rad/s²
6. None of the above.
7. Cannot be determined.
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accessible website, in whole or in part.
Assessing to Learn (4 of 11)
A 100 kg crate is attached to a rope wrapped around the inner disk as shown.
A person pulls on another rope wrapped around the outer disk with force F to
lift the crate. What force F is needed to lift the crate 2 m?
1.
2.
3.
4.
about 20 N
about 50 N
about 100 N
about 200 N
5.
6.
7.
8.
about 500 N
about 1,000 N
about 2,000 N
about 5,000 N
9. Impossible to determine without knowing the radii
10. Impossible to determine for some other reason(s)
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accessible website, in whole or in part.
Assessing to Learn (5 of 11)
A 100-kg crate is attached to a rope wrapped around the inner disk as shown.
A person pulls on another rope wrapped around the outer disk with force F to
lift the crate. What work W is needed to lift the crate 2 m?
1. about 400 J
2. slightly less than 2,000 J
3. Exactly 2,000 J
4. slightly more than 2,000 J
5. much more than 2,000 J
6. Impossible to determine without knowing F
7. Impossible to determine without knowing the radii
8. Impossible to determine without knowing the mass of the pulley
9. Impossible to determine for two or more of the reasons given in 6, 7, and 8 above
10. Impossible to determine for some other reason(s)
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accessible website, in whole or in part.
Assessing to Learn (6 of 11)
A spool has string wrapped around its center axle and is sitting on a horizontal
surface. If the string is pulled in the horizontal direction when tangent to the
top of the axle, the spool will...
1. ... roll to the right.
2. ... not roll, only slide to the right.
3. ... spin and slip, without moving left or right.
4. ... roll to the left.
5. None of the above
6. The motion cannot be determined.
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accessible website, in whole or in part.
Assessing to Learn (7 of 11)
A spool has string wrapped around its center axle and is sitting on a horizontal
surface. If the string is pulled in the horizontal direction when tangent to the
bottom of the axle, the spool will...
1. ... roll to the right.
2. ... not roll, only slide to the right.
3. ... spin and slip, without moving left or right.
4. ... roll to the left.
5. None of the above
6. The motion cannot be determined.
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accessible website, in whole or in part.
Assessing to Learn (8 of 11)
A spool has string wrapped around its center axle and is sitting on a horizontal
surface. If the string is pulled at an angle to the horizontal when drawn from
the bottom of the axle, the spool will...
1. ... roll to the right.
2. ... not roll, only slide to the right.
3. ... spin and slip, without moving left or right.
4. ... roll to the left.
5. None of the above
6. The motion cannot be determined.
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accessible website, in whole or in part.
Assessing to Learn (9 of 11)
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accessible website, in whole or in part.
Assessing to Learn (10 of 11)
Two blocks hang from strings wound around different parts of a double pulley as
shown. Assuming the system is not in equilibrium, what happens to the system's
potential energy when it is released from rest?
1.
It remains the same.
2.
3.
It decreases.
It increases.
4.
Impossible to determine without knowing the radii of the two pulleys.
5.
Impossible to determine without knowing the ratio of the radii of the two pulleys.
6.
Impossible to determine for some other reason.
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accessible website, in whole or in part.
Assessing to Learn (11 of 11)
Two blocks hang from strings would around different parts of a 2 kg double
pulley as shown. The pivot exerts a normal force FN supporting the double
pulley. Assuming the system is not in equilibrium, which statement about FN is
true after the system is released from rest? (Use g = 10 N/kg.)
1. FN = 20 N
2. 20 N < FN < 27 N
3. FN = 27 N
4. FN > 27N
5. It is impossible to predict what the normal force on
the double pulley will be.
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accessible website, in whole or in part.