Rotation Created by Craig Smiley (Harrison HS, West Lafayette, IN) Supported by grant PHY-0851826 from the National Science Foundation and by Purdue University

Rotational Motion (Fixed Axis)

Angular Position You are on a rotating merry-go-round. Based upon the pictures, what is your initial angular position form the +x-axis in radians? Counter-Clockwise is the positive direction. θ= π 4 θ= 3π 4 θ= π 2 θ=− π 2 θ=R Initial Final

Angular Displacement You are on a rotating merry-go-round. Based upon the pictures, what is your angular displacement? Counter- Clockwise is the positive direction. ∆θ= π 4 ∆θ= 3π 4 ∆θ= π 2 ∆θ=− π 2 ∆θ=R Can’t be determined without a reference point Initial Final

Angular Displacement You are on a rotating merry-go-round that goes 2 times around. What is your angular displacement? Counter- Clockwise is the positive direction. ∆θ=0 ∆θ=4π ∆θ=−4π ∆θ=4πR ∆θ=−4πR Initial Final

Angular Displacement You are on a rotating merry-go-round. Based upon the pictures, what is the direction of your angular displacement? Initial A. B. C. D. E. F. Final

θ vs t Which object had the greatest angular displacement? Purple Blue Angular Position Which object had the greatest angular displacement? Purple Blue Green Not enough information time

θ vs t What does it mean when the lines intersect on a angular position vs time graph? They have the same angular position They have the same angular displacement They have the same angular speed Both A & B A, B & C are true time Angular Position

θ vs t Which object is spinning faster? Orange Blue The same Angular Position Which object is spinning faster? Orange Blue The same Not enough information time

θ vs t Which is the correct description of the motion time Angular Position θ vs t Which is the correct description of the motion needed to make this angular position vs time graph? Rotate back and forth Initially rotating quickly in – direction, then abruptly rotates quickly in the + direction Initially rotating slowly in the – direction, then speed up in the + direction Initially rotating quickly in the – direction, then slows down, stop and then increases rotational speed in the + direction Initially rotating quickly in the – direction, then slows down, stop and then increases rotational speed still in the – direction

ω vs t Which is the correct description of the motion needed to make this angular velocity vs time graph? Spinning at a constant speed in + direction Spinning at a constant speed first in the – direction and then in the + direction Speeding up in the – direction and then turning around and speeding up in + direction Speeding up in the + direction the whole time Slowing down in the – direction and then turning around and speeding up in + direction Velocity time

Which difference in the rotational motion of the purple top compared to the blue top as shown on this angular velocity vs time graph? The purple top is spinning faster than the blue top and they are rotating in opposite directions The purple top is spinning faster than the blue top and they are rotating in the same direction The purple top is slowing down faster than the blue top is speeding up and they are going in opposite directions The purple top is slowing down faster than the blue top is speeding up and they are going in the same direction time Velocity

θ vs t  ω vs t Position Given this angular position vs time graph (red), which angular velocity vs time graph (blue) depicts the same motion? A. B. C. D. E. time Velocity Velocity time time time time time Velocity Velocity Velocity

θ vs t  ω vs t Given this angular velocity vs time graph (blue), which angular position vs time graph (red) depicts the same motion? A. B. C. D. E. Velocity time Position Position time time Position Position Position time time time

ω vs t  θ vs t Given this angular velocity vs time graph (blue), which angular position vs time graph (red) depicts the same motion? A. B. C. D. E. Velocity time Position Position time time Position Position Position time time time

ω vs t  θ vs t Given this angular velocity vs time graph (blue), which angular position vs time graph (red) depicts the same motion? A. B. C. D. E. Velocity time Position Position time time Position Position Position time time time

ω vs t A) ¾ B) 12 C) 24 D) 48 E) None of These F) Cannot Determine Given the angular velocity versus time graph to the right, what is the total angular displacement of the object? A) ¾ B) 12 C) 24 D) 48 E) None of These F) Cannot Determine 8 6 4 2 Velocity (m/s) t (s) 2 4 6 8 10 12 14 16 -2 -4 -6 -8

ω vs t A) 0 rad B) 18 rad C) 36 rad D) None of These Given the angular velocity versus time graph to the right, what is the total angular displacement of the object? A) 0 rad B) 18 rad C) 36 rad D) None of These E) Cannot Determine 8 6 4 2 Velocity (m/s) t (s) 2 4 6 8 10 12 14 16 -2 -4 -6 -8

ω vs t  α vs t Velocity Given this angular velocity vs time graph (blue), which angular acceleration vs time graph(green) depicts the same motion? A. B. C. D. E. time acceleration acceleration time time acceleration acceleration acceleration time time time

ω vs t  α vs t Velocity time Given this angular velocity vs time graph (blue), which angular acceleration vs time graph(green) depicts the same motion? A. B. C. D. E. acceleration acceleration time time acceleration acceleration time acceleration time time

θ vs t  α vs t Given this angular position vs time graph (red), what is the direction of angular acceleration? + – No direction Not enough information Position time

θ vs t  α vs t Position Given this angular position vs time graph (red), what is the direction of angular acceleration? + – No direction Not enough information time

ω vs t  α vs t Velocity Given this angular velocity vs time graph (blue), which angular acceleration vs time graph(green) depicts the same motion? A. B. C. D. E. time acceleration acceleration time time acceleration acceleration acceleration time time time

α vs t Given the angular acceleration versus time graph to the right, what is the change in angular velocity of the object? A) 2/5 rad/s B) 20 rad/s C) 40 rad/s D) None of these E) Cannot Determine 8 6 4 Acceleration (m/s2) 2 t (s) 2 4 6 8 10 12 14 16 -2 -4 -6 -8

Acceleration A merry-go-round has an acceleration of –3m/s2 and Ferris wheel has a acceleration of +2m/s2. Which of these statements MUST be true? They are spinning in opposite directions The merry-go-round is spinning faster than the Ferris wheel Both A & B The merry-go-round is slowing down and the Ferris wheel is speeding up. None of these have to be true

The bar is rotating at a constant rate. Which ball has a greater ω? Blue Green Both = 0 Both are equal ≠ 0 Can’t be determined

The bar is rotating at a constant rate. Which ball has a greater speed? Blue Green Both = 0 Both are equal ≠ 0 Can’t be determined

The bar is rotating at a constant rate. Which ball has a greater angular acceleration, α? Blue Green Both = 0 Both are equal ≠ 0 Can’t be determined

The bar is rotating at a constant rate. Which ball has a greater tangential acceleration, at? Blue Green Both = 0 Both are equal ≠ 0 Can’t be determined

The bar is rotating at a constant rate. Which ball has a greater centripetal acceleration, ac? Blue Green Both = 0 Both are equal ≠ 0 Can’t be determined

Moment of Inertia

Rotational Inertia (AKA Moment of Inertia) Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Ring = m Mass of disk = m  r   r 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Rod = m Mass of Rod = m  L   L 

Rotational Inertia (AKA Moment of Inertia) Which has a greater Rotational Inertia? A. B. C. They have the same D. Can’t be determined Mass of Disk = m Mass of Cylinder = m  r   r 

Rotational Inertia Which has a greater rotational inertia? (assume the balls to be point masses and the rods have negligible mass) A. B. C. They have the same D. Can’t be determined 2m m m  L   2L 

Rotational Inertia Which has a greater rotational inertia? (assume the balls to be point masses and the rods have negligible mass) A. B. C. They have the same D. Can’t be determined Mass of Ring = 2m m m  2L   L   L 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Ring = m Mass of Rod = m  2L   L 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Rod = m Mass of Rod = m  2L   L 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of Rod = m Mass of piece of pie = m  L   L 

Rotational Inertia Which has a greater rotational inertia? A. B. C. They have the same D. Can’t be determined Mass of piece of pie = m Mass of disk = m  2L   L 

Rotational Inertia Let m = 4kg, L = 2m, R = 0.5m Calculate the rotational inertia of both objects. Assume the rods have negligible mass. Point mass with mass m  L  m R  L 

Rotational Inertia (AKA Moment of Inertia) Which would have a greater Rotational Kinetic Energy? Both scenarios have a total of 8 tennis balls, all with a mass m, being spun at a constant rotational speed of ω, in a circle by length of string R. A. B. C. Both the same D. Can’t be determined

Torque

A 1 kg wheel with a fixed hub start from rest, and a force is applied as shown. Assume that the hub and spokes are massless, and that F = 1 N, and the momentum of inertia is I = MR2. What is the magnitude of the wheel’s angular acceleration? M = 1 kg |F| = 1N R = 0.5 m Q = 60o w/r to horizontal ¼ rad/sec2 ½ rad/sec2 1 rad/sec2 2 rad/sec2 4 rad/sec2

A wheel (mass of 1kg and with a radius of 1m) with a fixed hub starts from rest, and forces are applied as shown. Assume that the hub and spokes are massless, and that the wheel is a hoop, I = MR2. F1 = 1 N and acts at a distance of half of the radius of the wheel. What is the angular acceleration of the wheel? F1=1N 1m 0.5m 0.25 rad/s2 0.50 rad/s2 1 rad/s2 2 rad/s2 4 rad/s2

Two identical wheels (mass of 1kg and with a radius of 1m) with fixed hubs start from rest, and forces are applied as shown. Assume that the hub and spokes are massless, and that F1 = 1 N. In order to impart identical angular accelerations, how large must F2 be? I = MR F1 F2 M = 1 kg R1 = 0.5 m R2 = 1.0 m 0.25 N 0.50 N 1 N 2 N 4 N F1 1m 0.5m

Two 1 kg wheels with fixed hubs start from rest, and forces are applied as shown. Assume that the hub and spokes are massless, and that F1 = 1 N. In order to impart identical angular accelerations, how large must F2 be? I = MR2 F1 F2 M = 1 kg R1 = 0.5 m R2 = 1.0 m 0.25 N 0.50 N 1 N 2 N 4 N

Rolling

Big Ball vs Small Ball Which will reach the bottom first? Big Small Tie

Big Hoop vs Small Hoop Which will reach the bottom first? Big Small Tie

Big Disk vs Small Disk Which will reach the bottom first? Big Small Tie

Solid Cylinder vs Disk Which will reach the bottom first? Cylinder Tie

Solid Cylinder vs Hollow Cylinder Which will reach the bottom first? Solid Hollow Tie

Solid Cylinder vs Solid Sphere Which will reach the bottom first? Cylinder Sphere Tie

Solid Sphere vs Rolling Cart Which will reach the bottom first? Sphere Cart Tie

Bowling Ball When a bowling ball first starts going down the bowling alley it is rolling AND SLIDING. At a given moment in time during this process, is 𝑣 𝑐𝑚 =𝜔𝑟 ? Yes No Possibly

Bowling Ball When a bowling ball first starts going down the bowling alley it is rolling AND SLIDING. During this time, is 𝑎 𝑐𝑚 =𝛼𝑟 ? Yes No Possibly

Bowling Ball When a bowling ball first starts going down the bowling alley it is rolling AND SLIDING. During this time, what is the sign for 𝑎 𝑐𝑚 and 𝛼 ? (Assume the positive direction to be the way the ball is moving and the way the ball is spinning) acm (+) α (+) acm (+) α (–) acm (–) α (+) acm (–) α (–) Can’t be determined

Falling Boards Two boards of equal length, but unequal masses, have one end raised so they angle up like an incline. Which board will hit the ground first? The less massive board The more massive board The same time Can’t be determined

Falling Boards Two boards of equal length, but unequal masses, have one end raised so they angle up like an incline. The more massive board has an additional mass attached to the raised end. Which board will hit the ground first? The less massive board The more massive board The same time Can’t be determined

Falling Boards Two boards of unequal length have one end raised so they angle up like an incline. Which board will hit the ground first? The shorter board The longer board The same time Can’t be determined

Angular Momentum

Which has more angular momentum if both objects have the same rotational velocity? A. B. C. The same D. Can’t be determined

Identical objects are spinning at the same speed, but in opposite directions. Which measurements are the same for both objects? Angular Momentum Rotational Kinetic Energy Both A & B Neither A nor B

What could you calculate using this L vs t graph to determine the Net Torque? Slope of the line Area between the line and the x-axis Y-intercept None of these

A merry-go-round initially spinning clockwise traveling experiences a positive applied torque from the uncle as shown in the graph. Clockwise torques are considered are positive. While traveling the full 0.9 sec, the merry-go-round’s rotational speed: Torque Time (seconds) 0 0.3 0.9 1000 mN 0 mN first increases and then decreases. first decreases and then increases. continuously increases. drops to zero at 0.9 sec. None of the above.

A merry-go-round initially spinning clockwise traveling experiences a positive applied torque from the uncle as shown in the graph. Clockwise torques are considered are positive. While traveling the full 0.9 sec, what is the merry-go-round’s change in angular momentum? Torque Time (seconds) 0 0.3 0.9 1000 mN 0 mN 0 kg(m2)/sec 10 kg(m2)/sec 300 kg(m2)/sec 450 kg(m2)/sec 900 kg(m2)/sec

Two disks—A and B—on frictionless axles are initially at rest Two disks—A and B—on frictionless axles are initially at rest. Disk A has twice the moment of inertia as Disk B. Now you exert twice as much Torque on Disk A as you do on Disk B, both for 1 second. One second later, which measurements of Disk A and Disk B are the same? Change in Angular Momentum Angular Acceleration Change in Angular Velocity A & B B & C A, B & C 2τ A τ B 68

A constant torque is exerted on a disk that is initially at rest on a frictionless axle. The torque acts for a short time interval and gives the disk a final speed. To reach the same speed using a torque that is half as big, the torque must be exerted for a time interval that is Four times as long. Twice as long. The same length. Half as long. A quarter as long. τ 69

Two disks—A and B—on frictionless axles are initially at rest Two disks—A and B—on frictionless axles are initially at rest. Disk A has twice the moment of inertia as Disk B. Now you exert the same constant torque on both disks for 1 second. One second later, the momentum of disk A is: Twice the angular momentum of Disk B The same as the angular momentum of Disk B Half the angular momentum of Disk B Not enough information to determine τ A τ B 70

Two identical disks, A and B, initially are spinning on frictionless axles. The initial rotational velocity of Disk A is twice as that of Disk B. You then exert the same constant torque on the two disks over 1 second. One second later, the change in angular momentum of Disk A is: 2ω Non-zero and twice the change in angular momentum of Disk B Non-zero and the same as the change in angular momentum of Disk B Zero. Non-zero and half the change in angular momentum of Disk B Not enough information to determine τ A ω τ B 71

A child (the blue dot) with is running at a constant velocity v A child (the blue dot) with is running at a constant velocity v. Which of these line segments would be used to calculate the child’s angular momentum about point P? D. A child running in a straight line can’t have angular momentum because they are not going in a circular path. P A B C

A child (the blue dot) with is running at a constant velocity v A child (the blue dot) with is running at a constant velocity v. What is the direction of the child’s angular momentum about point P? Clockwise Counter-Clockwise Can’t be determined P

Into Out of the the page page Left Right rod The non-spinning axle of a rapidly spinning wheel is attached to a non-spinning rod supported by a pole on one end. The direction of spin is such that a point on the wheel is coming at you when it is at the bottom of the wheel and moving away from you when it is at the top. What is the direction of the wheel’s angular momentum? (Use the right-hand rule) Into Out of the the page page Left Right rod wheel pole Into the page Out of the page Up (on the page) Down (on the page) Left (on the page) Right (on the page)

Into Out of the the page page Left Right rod The force of gravity on the wheel plus rod is downward. What is the direction of the resulting torque on the wheel-plus-rod system about the pointed support? (The wheel is spinning very rapidly.) Into Out of the the page page Left Right rod wheel mg pole Into the page. Out of the page Up (on the page) Down (on the page) Left (on the page) Right (on the page)

Because of the gravitational force/torque, the wheel-plus-rod system begins to move. Viewed from above, what does it do? rod wheel mg Left Right rod wheel mg pole Fall down Rotate clockwise Rotate counterclockwise Not enough information to determine