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 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 Betsy’s 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 f=159 rpm

What is a Radian? The “radian pie” has an arc equal to the radius 2  radians =  radians = 1 revolution

Angular Velocity Angular Velocity = radians / time  =  / t

 and f rad / s = (2  ) rev/s  = 2  f

 and v v = 2  f r and  = 2  f so … v = r 

Example 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 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/s 2 ) Angular (rad/s 2 ) a  ( v f - v i ) / t (  f -  i ) / t a = r 

Example 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

Solution 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 Awesome Angular Analogies d = v i t + 1/2 a t 2 ?

Solution Awesome Angular Analogies d = v i t + 1/2 a t 2  =  i t + 1/2  t 2

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

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

Example Moment of Inertia Experiment The same force is applied to all. Which one will undergo the greatest angular acceleration?

Solution 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 = mr 2 The moment of inertia of a particle of mass m spinning at a distance r is I = mr 2 For the same torque, the smaller the moment of inertia, the greater the angular acceleration  = I 

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Example Sarah Hughes Will her mass change when she pulls her arms in? Will her moment of inertia change?

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

Example 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

Solution 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 = MR 2 I = 1/2 MR 2 I = 2 /5 MR 2 hoop disc ball

Example K.E. of Rotation What is the formula for the kinetic energy of rotation? A. 1/2 mv 2 B. 1/2 m  2 C. 1/2 I  2 D. I 

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

Solution 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 I 1  1 = I 2  2

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Example 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 Sarah Hughes B. When her arms stretch out her moment of inertia increases and her angular velocity decreases I 1  1 = I 2  2 So when I increases,  decreases!

That’s all folks!