Ten sense-making tasks on rotational kinematics. Explain your reasoning for every answer. The explanation counts as much as the answer.
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Six disks each spin at a constant angular speed $\omega$ about a fixed axle through their centers. The radius $r$ and angular speed of each disk are given. The red dot is a point on the rim of each disk. (Disks are drawn to scale.)
Rank the disks by the linear (tangential) speed of the red dot, from greatest to least. If any are equal, show the tie. If the ranking cannot be determined, say so.
Greatest 1 ____ 2 ____ 3 ____ 4 ____ 5 ____ 6 ____ Least
Explain your reasoning. Would your ranking change if you ranked by angular speed instead?
Back to topMaya sits close to the center of a merry-go-round and Leo sits at the outer edge. The merry-go-round turns counterclockwise at a constant rate. (Top view.)
Four students discuss the two riders:
Ava: "Leo has the larger angular speed, because he travels around a much bigger circle in the same time."
Ben: "They have the same angular speed, but Leo has the larger linear speed."
Cam: "They go around once in the same amount of time, so they have the same linear speed too."
Dee: "If the merry-go-round starts speeding up, Leo will have a bigger angular acceleration than Maya."
Which student(s), if any, do you agree with? Explain what is right or wrong about each statement.
Back to topThe graph shows the angular velocity of a wheel. Counterclockwise is positive.
Student explanation: "At $t = 3\ \text{s}$ the wheel stops, so its angular acceleration is zero at that instant. After $3\ \text{s}$ the wheel spins the other way, so its angular acceleration changes from negative to positive."
What, if anything, is wrong with this explanation? If something is wrong, explain the error and give the correct description of the angular acceleration. If nothing is wrong, explain why it is correct.
Back to topA wheel rotates about a fixed axle. Counterclockwise is the positive direction.
For each case, choose what is happening to the wheel's angular speed at that instant:
A. speeding up B. slowing down C. constant angular speed D. cannot be determined
| Case | Choice |
|---|---|
| (a) $\omega = +4\ \text{rad/s}$, $\alpha = +2\ \text{rad/s}^2$ | |
| (b) $\omega = +4\ \text{rad/s}$, $\alpha = -2\ \text{rad/s}^2$ | |
| (c) $\omega = -4\ \text{rad/s}$, $\alpha = -2\ \text{rad/s}^2$ | |
| (d) $\omega = -4\ \text{rad/s}$, $\alpha = +2\ \text{rad/s}^2$ | |
| (e) $\omega = -4\ \text{rad/s}$, $\alpha = 0\ \text{rad/s}^2$ | |
| (f) $\alpha = -2\ \text{rad/s}^2$ (no information about $\omega$) |
Explain the rule you used. What do the cases with the same answer have in common?
Back to topTwo bikes ride side by side at the same constant speed of 6.0 m/s. Bike A has wheels of radius 0.20 m. Bike B has wheels of radius 0.40 m. All wheels roll without slipping.
For each quantity, state whether it is greater for A, greater for B, or the same for both. If different, give the ratio.
| Quantity | Answer |
|---|---|
| (a) Speed of the axle (center of the wheel) | |
| (b) Angular speed of the wheel | |
| (c) Number of wheel rotations needed to travel 120 m | |
| (d) Speed of the point of the tire touching the ground, relative to the ground |
Explain each answer, and calculate the angular speed of each wheel.
Back to topA turntable starts from rest and speeds up with a constant angular acceleration of $1.5\ \text{rad/s}^2$. A sticker is on its edge.
The equation below describes the rotation of some object:
$$\Delta\theta = (2.0\ \text{rad/s})(4.0\ \text{s}) + \tfrac12\,(3.0\ \text{rad/s}^2)(4.0\ \text{s})^2$$
The graph shows the angular velocity of a spinning wheel. Counterclockwise is positive.
A belt connects a small pulley (radius 0.10 m) to a large pulley (radius 0.30 m). A motor turns the small pulley at 30 rad/s. The belt does not slip.
Student solution:
The belt connects the pulleys, so they turn together: $\omega_{large} = \omega_{small} = 30\ \text{rad/s}$.
Speed of the large pulley's rim: $v = r\omega = (0.30\ \text{m})(30\ \text{rad/s}) = 9.0\ \text{m/s}$.
So the belt moves at 9.0 m/s.
Identify the error in the solution. Explain the physics of why it's an error, then find the correct angular speed of the large pulley and the speed of the belt.
Back to topA bowler releases a ball of radius 0.10 m so that it slides down the lane at 8.0 m/s without spinning at all. For the first several meters the ball skids. Friction from the lane gradually makes it start to spin, until eventually it rolls without slipping.
Riley writes: "Just after release, the ball's angular speed is $\omega = \dfrac{v}{R} = \dfrac{8.0\ \text{m/s}}{0.10\ \text{m}} = 80\ \text{rad/s}$."