Physics 121

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

Example 8.1 . . . Betsy’s new bike

The radius of the wheel is 30 cm and the speed v = 5 m/s. What is the rpm (revolutions per

minute) ?

Solution 8.1 . . . Betsy’s new bike

r = radiuscircumference = 2 rf = revolutions per secondv = d/t

v = 2 f r

5 = (2 )(f)(0.3)f = 2.6 revolutions per second f=159 rpm

What is a Radian?

The “radian pie” has an arc equal to the radius

2 radians = 3600

2 radians = 1 revolution

Angular Velocity

Angular Velocity = radians / time

= / t

and f

rad / s = (2 ) rev/s

= 2 f

and v

v = 2 f rand

= 2 fso …

v = r

Example 8.2 . . . Betsy’s

The radius of the wheel is 30 cm. and the (linear) velocity, v, is 5 m/s. What is Betsy’s

angular velocity?

Solution 8.2 . . . Betsy’s

v = r 5 = (0.3)()

= 16.3 rad/s

v and

Linear (m/s) Angular (rad/s)

v

d / t / t

2 r f 2 f

v = r

a and

Linear (m/s2) Angular (rad/s2)

a ( vf - vi ) / t ( f - i ) / t

a = r

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. moreB. lessC. same

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!

Angular Analogs

d

v

a

Example 8.4 . . . Awesome Angular Analogies

d = vi t + 1/2 a t2 ?

Solution 8.4 . . . Awesome Angular Analogies

d = vi t + 1/2 a t2 = i t + 1/2 t2

Torque

Torque means the “turning effect” of a force

SAME force applied to both. Which one will turn easier?

Torque

Torque = distance x force

= r x F

Easy!

Torque

Which one is easier to turn now?

Torque . . . The Rest of the Story!

= r F sin

Easy!

Example 8.5 . . . Inertia Experiment

The same force is applied to m and M. Which one accelerates more?

Solution 8.5 . . . Inertia Experiment

Since F = ma, the smaller mass (m) will accelerate more.

Example 8.6 . . . Moment of Inertia Experiment

The same force is applied to all. Which one will undergo the greatest angular acceleration?

Solution 8.6 . . . Moment of Inertia Experiment

This one will undergo the greatest angular acceleration.

What is Moment of Inertia?

F = m a

Force = mass x ( linear ) acceleration

= I

Torque = moment of inertia x angular acceleration

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

All about Sarah Hughes . . .

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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?

Solution 8.7 . . . Sarah Hughes

Mass does not change when she pulls her arms in but her moment of inertia decreases.

Example 8.8 . . . Guessing Game

A ball, hoop, and disc have the same mass. Arrange in order of decreasing I

A. hoop, disc, ballB. hoop, ball, discC. ball, disc, hoopD. disc, hoop, ball

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

Example 8.9 . . . K.E. of Rotation

What is the formula for the kinetic energy of rotation?

A. 1/2 mv2

B. 1/2 m2

C. 1/2 I2

D. I

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

Example 8.10 . . . Angular Momentum

Guesstimate the formula for angular momentum?

A. mvB. mC. I D. 1/2 I

Solution 8.10 . . . Angular Momentum

• Guesstimate the formula for the angular momentum?

• Linear Momentum is mv

• Angular Momentum is I

Conservation of Angular Momentum

• In the absence of any external torques, the angular momentum is conserved.

• If = 0 then I11 = I2 2

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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

Solution 8.11 . . . Sarah Hughes

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

• I11 = I2 2

• So when I increases, decreases!

That’s all folks!