Physics 121

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Physics 121
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8. Rotational Motion
8.1 Angular Quantities 8.2 Kinematic
Equations 8.3 Rolling Motion 8.4 Torque 8.5
Rotational Inertia 8.6 Problem Solving
Techniques 8.7 Rotational Kinetic Energy 8.8
Conservation of Angular Momentum
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Example 8.1 . . . Betsys new bike
The radius of the wheel is 30 cm and the speed v
5 m/s. What is the rpm (revolutions per
minute) ?
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Solution 8.1 . . . Betsys new bike
r radius circumference 2 ? r f revolutions
per second v d/t v 2 ? f r 5 (2
?)(f)(0.3) f 2.6 revolutions per second f159
rpm
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What is a Radian?
The radian pie has an arc equal to the radius
2 ? radians 3600 2 ? radians 1 revolution
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Angular Velocity
Angular Velocity radians / time ? ? / t
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? and f
rad / s (2 ?) rev/s ? 2 ?f
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? and v
v 2 ? f r and ? 2 ?f so v r ?
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Example 8.2 . . . Betsys ?
The radius of the wheel is 30 cm. and the
(linear) velocity, v, is 5 m/s. What is Betsys
angular velocity?
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Solution 8.2 . . . Betsys ?
v r ? 5 (0.3)(?) ? 16.3 rad/s
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v and ?
Linear (m/s)
Angular (rad/s) v
? d / t
? /
t 2 ? r f
2 ? f v r ?
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a and ?

• Linear (m/s2)
  Angular (rad/s2)
• a
  ?
• ( vf - vi ) / t
  ( ?f - ?i ) / t
• a r ?

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Example 8.3 . . . CD Music
To make the music play at a uniform rate, it is
necessary to spin the CD at a constant linear
velocity (CLV). Compared to the angular velocity
of the CD when playing a song on the inner track,
the angular velocity when playing a song on the
outer track is A. more B. less C. same
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Solution 8.3 . . . CD Music
v r ? When r increases, ? must decrease in
order for v to stay constant. Correct choice is
B Note Think of track races. Runners on the
outside track travel a greater distance for the
same number of revolutions!
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Angular Analogs

• d
  ?
• v
  ?
• a
  ?

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Example 8.4 . . . Awesome Angular Analogies

• d vi t 1/2 a t2
  ?

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Solution 8.4 . . . Awesome Angular Analogies

• d vi t 1/2 a t2
  ? ?i t 1/2 ? t2

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Torque
Torque means the turning effect of a
force SAME force applied to both. Which one
will turn easier?
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Torque

• Torque distance x force
• ? r x F

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Torque

• Which one is easier to turn now?

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Torque . . . The Rest of the Story!

• ? r F sin ?

Easy!
?
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Example 8.5 . . . Inertia Experiment
The same force is applied to m and M. Which one
accelerates more?
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Solution 8.5 . . . Inertia Experiment
Since F ma, the smaller mass (m) will
accelerate more.
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Example 8.6 . . . Moment of Inertia Experiment
The same force is applied to all. Which one will
undergo the greatest angular acceleration?
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Solution 8.6 . . . Moment of Inertia Experiment
This one will undergo the greatest angular
acceleration.
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What is Moment of Inertia?

• F m a
• Force mass x ( linear ) acceleration
• ? I ?
• Torque moment of inertia x angular
  acceleration

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I mr2

• The moment of inertia of a particle of mass m
  spinning at a distance r is
• I mr2
• For the same torque, the smaller the moment of
  inertia, the greater the angular acceleration
• ? I ?

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All about Sarah Hughes . . .

• Click me!

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Example 8.7 . . . Sarah Hughes

• Will her mass change when she pulls her arms in?
• Will her moment of inertia change?

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Solution 8.7 . . . Sarah Hughes
Mass does not change when she pulls her arms in
but her moment of inertia decreases.
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Example 8.8 . . . Guessing Game
A ball, hoop, and disc have the same mass.
Arrange in order of decreasing I A. hoop, disc,
ball B. hoop, ball, disc C. ball, disc, hoop D.
disc, hoop, ball
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Solution 8.8 . . . Guessing Game
A. hoop, disc, ball I (moment of inertia)
depends on the distribution of mass. The farther
the mass is from the axis of rotation, the
greater is the moment of inertia. I MR2
I 1/2 MR2 I 2 /5
MR2 hoop disc
ball
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Example 8.9 . . . K.E. of Rotation
What is the formula for the kinetic energy of
rotation? A. 1/2 mv2 B. 1/2 m?2 C. 1/2 I?2 D. I
?
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Solution 8.9 . . . K.E. of Rotation

• The analog of v is ?
• The analog of m is I
• The K.E. of rotation is 1/2 I ?2

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Example 8.10 . . . Angular Momentum
Guesstimate the formula for angular momentum? A.
mv B. m? C. I ? D. 1/2 I ?
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Solution 8.10 . . . Angular Momentum

• Guesstimate the formula for the angular momentum?
• Linear Momentum is mv
• Angular Momentum is I ?

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Conservation of Angular Momentum

• In the absence of any external torques, the
  angular momentum is conserved.
• If ? ? 0 then I1?1 I2 ?2

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More about Sarah Hughes . . .

• Click me!

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Example 8.11 . . . Sarah Hughes

• A. When her arms stretch out her moment of
  inertia decreases and her angular velocity
  increases
• B. When her arms stretch out her moment of
  inertia increases and her angular velocity
  decreases
• C. When her arms stretch out her moment of
  inertia decreases and her angular velocity
  decreases
• D. When her arms stretch out her moment of
  inertia increases and her angular velocity
  increases

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Solution 8.11 . . . Sarah Hughes

• B. When her arms stretch out her moment of
  inertia increases and her angular velocity
  decreases
• I1?1 I2 ?2
• So when I increases, ? decreases!

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Thats all folks!