Ten sense-making tasks on rotational dynamics, rotational energy, and angular momentum. Explain your reasoning for every answer. The explanation counts as much as the answer.
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Each object starts at rest and can spin freely about a fixed, frictionless axle through its center. A single force is applied tangentially at the rim. Use $I_{disk} = \tfrac12 MR^2$ and $I_{hoop} = MR^2$. (Objects are drawn to scale.)
Rank the objects by the magnitude of their angular acceleration, 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.
Back to topA large, heavy hoop (5.0 kg, radius 0.30 m) and a small, light solid disk (1.0 kg, radius 0.10 m) are released from rest side by side at the top of identical ramps. Each one drops a vertical height of 1.2 m to the bottom. Both roll without slipping.
Three students predict the winner:
Ava: "The hoop wins. It's five times heavier, so gravity pulls it down the ramp much harder."
Ben: "The hoop loses because it's bigger. A bigger radius means it has to turn more, which slows it down."
Cam: "The disk wins, and its mass and radius don't matter. Any solid disk beats any hoop because the disk puts a smaller fraction of its energy into spinning."
Which student(s), if any, do you agree with? Explain what is right or wrong about each statement. Then find each object's speed at the bottom.
Back to topA figure skater spins on frictionless ice with her arms out. She then pulls her arms in.
Student explanation: "No external torque acts, so angular momentum is conserved: $(4.0\ \text{kg·m}^2)(2.0\ \text{rad/s}) = (1.6\ \text{kg·m}^2)\,\omega$, so $\omega = 5.0\ \text{rad/s}$. Since nothing outside her does any work, her rotational kinetic energy is conserved too, so it stays the same."
What, if anything, is wrong with this explanation? If something is wrong, identify it, explain the error, and give the correct description. If nothing is wrong, explain why it is correct.
Back to topFor each situation, decide what happens to the angular momentum of the stated system about the rotation axis during the event described:
A. increases B. decreases C. stays the same D. cannot be determined
| Situation (system) | Choice |
|---|---|
| (a) A spinning skater on frictionless ice pulls in her arms. (System: skater) | |
| (b) A lump of clay is dropped straight down onto a freely spinning turntable and sticks. (System: turntable + clay) | |
| (c) A motor speeds up a ceiling fan. (System: fan blades) | |
| (d) Friction in the axle gradually slows a spinning bicycle wheel. (System: wheel) | |
| (e) A child runs tangent to the edge of a stationary merry-go-round and jumps on. The axle is frictionless. (System: child + merry-go-round) | |
| (f) The same event as (e). (System: merry-go-round only) | |
| (g) A wheel experiences a net external torque, but its direction compared with the wheel's rotation is not known. (System: wheel) |
Explain the rule you used. In (b), what happens to the turntable's angular speed?
Back to topA solid disk and a hoop each have mass 2.0 kg and radius 0.50 m, and both start at rest on frictionless axles. Each gets a 4.0 N force applied tangentially at its rim for 3.0 s.
For each quantity at the end of the 3.0 s, state whether it is greater for the disk, greater for the hoop, or the same for both. If different, give the ratio.
| Quantity | Answer |
|---|---|
| (a) Angular impulse delivered | |
| (b) Final angular momentum | |
| (c) Final angular speed | |
| (d) Final rotational kinetic energy |
Explain each answer. If both objects got the same angular impulse, how can their kinetic energies be different?
Back to topA 1.0 kg block hangs from a light string wrapped around a pulley that is a solid disk (mass 2.0 kg, radius 0.20 m) on a frictionless fixed axle. The block is released from rest. The string does not slip.
The equation below is an energy equation for some physical situation:
$$(3.0\ \text{kg})(10\ \text{m/s}^2)(1.2\ \text{m}) = \tfrac12(3.0\ \text{kg})\,v^2 + \tfrac12\Big[\tfrac12(3.0\ \text{kg})(0.10\ \text{m})^2\Big]\left(\frac{v}{0.10\ \text{m}}\right)^2$$
Three objects of equal mass start from rest at the same height on identical ramps: a solid cylinder that rolls without slipping, a hoop that rolls without slipping, and a block that slides down a frictionless ramp. Take the bottom of the ramp as the zero of gravitational potential energy. At the top, each object's energy is shown below (6 units).
A disk with rotational inertia $0.45\ \text{kg·m}^2$ and radius 0.50 m is at rest on a frictionless axle. A 0.20 kg lump of clay moving at 5.0 m/s, tangent to the rim, hits the rim and sticks. (Top view.)
Student solution:
Energy is conserved, so the clay's kinetic energy becomes the disk's rotational kinetic energy:
$\tfrac12(0.20\ \text{kg})(5.0\ \text{m/s})^2 = \tfrac12(0.50\ \text{kg·m}^2)\,\omega^2$, using $I_{total} = 0.45\ \text{kg·m}^2 + (0.20\ \text{kg})(0.50\ \text{m})^2 = 0.50\ \text{kg·m}^2$
$2.5\ \text{J} = (0.25\ \text{kg·m}^2)\,\omega^2 \;\Rightarrow\; \omega = 3.2\ \text{rad/s}$
Identify the error in the solution. Explain the physics of why it's an error, then find the correct angular speed. How much kinetic energy is actually converted to other forms?
Back to topA 0.50 kg hockey puck slides in a straight line at a constant 4.0 m/s on frictionless ice. Its path passes a post; the closest distance between the path and the post is 0.30 m. (Top view.)
Jada writes: "The angular momentum of the puck about the post is $L = mvr_\perp = (0.50\ \text{kg})(4.0\ \text{m/s})(0.30\ \text{m}) = 0.60\ \text{kg·m}^2/\text{s}$."
Theo replies: "That's meaningless. The puck isn't rotating or moving in a circle, so it has no angular momentum."