Two wheels lock together: conserve L, track the energy
You will be able to: Predict a shared angular velocity and account for kinetic-energy loss in a rotational collision.
What happens when a stationary disk is coupled to a spinning disk?
A spinning disk touches a stationary coaxial disk. Friction brings them to one shared angular velocity. With negligible external torque about the axle, the pair retains its total angular momentum, while some kinetic energy becomes thermal energy.
A useful starting point: Pull inward: faster spin, unchanged angular momentum →
Words and symbols before equations
- Coaxial
- Sharing the same rotation axis.
- Coupling
- An interaction that makes the disks eventually rotate together.
- Inelastic rotational interaction
- An interaction in which the disks share a final ω and rotational kinetic energy generally decreases.
What this picture assumes
Coaxial disks: I_1=2 kg·m², ω_1=+6 rad/s; I_2=1 kg·m². They lock with negligible external angular impulse. CCW positive. Kinetic-energy decrease becomes internal energy.
Connect the picture to the physics
For the two-disk system, internal friction transfers angular momentum between the disks. Neglect external torque during coupling: I_1ω_1+I_2ω_2=(I_1+I_2)ω_f. Include negative angular velocities for opposite initial rotations.
Calculate kinetic energy separately before and after using ½Iω². The decrease becomes thermal energy and possibly sound or deformation. Friction need not destroy angular momentum of the combined system; its internal torques cancel in the angular-momentum balance.
An experimental test can measure both disks’ inertias and their angular velocities immediately before and after coupling, then compare signed total L with uncertainties. Measure free spin-down to estimate axle-friction effects. Repeat trials; agreement within uncertainty supports the approximation without proving exact zero external torque.
A worked example, step by step
Disk 1 has I_1=2 kg·m² and ω_1=6 rad/s. Disk 2 has I_2=1 kg·m² and starts at rest. They lock together with negligible external torque. Find ω_f and energy loss.
- L_i=2(6)+1(0)=12 kg·m²/s.
- Combined inertia=3 kg·m², so ω_f=12/3=4 rad/s.
- K_i=½(2)(6²)=36 J; K_f=½(3)(4²)=24 J.
- 12 J of rotational kinetic energy becomes other energy forms. Angular momentum remains 12 kg·m²/s.
Do not use kinetic-energy conservation for disks that rub and lock together.
When would ideal coupling cause no kinetic-energy decrease?
Compare with an explanation
When both disks already have the same angular velocity, so there is no relative rotation to remove.
Predict. Change one thing. Explain.
Vary the second disk’s initial angular velocity, including counterrotation. Find a case with zero final spin and nonzero initial kinetic energy. Then identify when no energy is lost.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Total L=12 kg·m²/s; shared ω_f=4 rad/s. K_i=36 J, K_f=24 J, decrease=12 J. External angular impulse is negligible.
Coaxial disks: I_1=2 kg·m², ω_1=+6 rad/s; I_2=1 kg·m². They lock with negligible external angular impulse. CCW positive. Kinetic-energy decrease becomes internal energy.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant physical relationship or contact condition to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionIdentical disks each have I=1 kg·m² and initial ω values +6 and −2 rad/s. They lock with negligible external torque. (a) Find initial total L. (b) Find shared ω_f. (c) Find K_i and K_f. (d) Propose one measurement to test the negligible-external-torque assumption.
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Compare with the answer and four-point rubric
- 1 point: L_i=6−2=4 kg·m²/s.
- 1 point: ω_f=4/2=2 rad/s.
- 1 point: K_i=18+2=20 J; K_f=½(2)(4)=4 J.
- 1 point: Measure spin-down without disk contact to estimate axle torque; compare its angular impulse over the coupling interval with measured L, allowing for measurement uncertainty.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1What law predicts shared ω in an isolated rotational collision?
Conservation of total angular momentum.
RECALL 2Does internal friction force total L to decrease?
No; its paired internal torques cancel.
RECALL 3What happens to lost rotational K?
It becomes thermal/internal energy and possibly sound.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Two wheels lock together: conserve L, track the energy
- ω_f=(I_1ω_1+I_2ω_2)/(I_1+I_2), negligible external impulse.
- Calculate K_i−K_f separately; it becomes thermal/internal energy.
Remember: Do not use kinetic-energy conservation for disks that rub and lock together.
Conditions: Coaxial disks: I_1=2 kg·m², ω_1=+6 rad/s; I_2=1 kg·m². They lock with negligible external angular impulse. CCW positive. Kinetic-energy decrease becomes internal energy.
Refresh Kid · Unit 6 · Objectives 6.4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.4, objectives 6.4.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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