Conserve angular momentum without assuming constant kinetic energy

1. Frame the question

For a system about the specified inertial reference, negligible net external torque makes total angular momentum constant. Choose the system large enough to include the internal exchange being studied. The torque one part exerts on another can change each part’s angular momentum while the opposite internal exchange cancels in the total.

2. Build the mechanism

A nonrigid system can change its mass distribution and angular speed while conserving angular momentum. Comparing initial and final uniformly rotating states gives Iiωi=Ifωf. If inertia falls, angular speed rises. This is not an application of constant-I τ=Iα across the redistribution: inertia is changing, and the full endpoint angular-momentum relation is needed.

Conserved angular momentum does not guarantee conserved rotational kinetic energy. Substituting ω=L/I gives K=L²/(2I). At fixed L, reducing inertia increases this energy; internal work and energy conversion supply it. The model compares endpoint states after radial motion has ended and does not pretend to describe the intermediate motion or effort.

A different case joins two coaxial rotors through internal friction with negligible external axle torque. The final common speed is (I1ω1+I2ω2)/(I1+I2). Relative rotation disappears and kinetic energy generally decreases, becoming internal energy. For I1=2, I2=1 kg m² and speeds +4 and −2 rad/s, total L=6, final speed 2 rad/s, initial energy 18 J and final 6 J. The 12 J decrease is not a loss of angular momentum.

3. Work an example

The redistribution model holds L=6 kg m²/s. With initial inertia 4 and final 2 kg m², angular speed changes from 1.5 to 3 rad/s. Rotational energy rises from 4.5 to 9 J, requiring 4.5 J conversion into rotational kinetic energy under the stated endpoint assumptions. Reversing the redistribution reverses that energy change.

4. Predict and investigate

Predict first: Can rotational kinetic energy increase with no external torque? What supplies the added energy?

Compare inertia changes 4→2, 2→4 and 3→3. Calculate both angular speeds and kinetic energies. Then solve the separate two-rotor joining case from the explanation and identify why its internal energy change has the opposite sign from contraction. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

An isolated system has L=8 kg m²/s and final inertia 2 kg m². Find final rotational kinetic energy.

Your numerical result (J) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

If only one of two interacting rotors is your system, is its angular momentum necessarily conserved? Explain the boundary rather than merely naming a law.

Retain your first response, system and axis choices, separate scalar energy and signed angular-momentum ledgers, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic captures are not a performed physical experiment.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 6 topic mapping: 6.4. Checked 30 October 2026. Planar signed analysis without required three-dimensional angular-momentum direction. Slipping evolution is qualitative, in accordance with the framework boundary; no rolling-resistance model. Framework cross-reference, not AP authorization.