Conserve angular momentum while inertia changes

Select the whole interacting system

About the fixed axis, total L is the sum of the angular momenta of all included components. Internal interactions transfer equal and opposite angular impulses between those components. If net external angular impulse is zero, total L is unchanged, even though a smaller selected subsystem can gain or lose angular momentum. Zero external torque is the relevant condition, not simply zero external force.

Account for redistribution

The virtual system consists of a central inertia 1 kg m² and two 1 kg point masses initially at radius 2 m. Initial inertia is 9 kg m². When both radii become r, If=1+2r². With initial angular velocity omega_i, L=9omega_i and omega_f=L/If. During redistribution d(I omega)/dt=I alpha+omega dI/dt; dropping the second term would falsely imply constant angular speed.

Find the energy source

At endpoints with zero radial speeds, K=L²/(2I). Smaller I therefore increases K at fixed L. Internal agents moving the masses supply the increase; angular-momentum conservation does not require conservation of rotational kinetic energy. The workbench is an endpoint model, not a prescription for the radial forces or transient radial kinetic energy.

Test a different interaction

Consider two coaxial rigid rotors that couple without external torque: I1=2 with omega1=3, and I2=1 with omega2=−2 in SI units. Total initial L=4, so common final omega=4/3. Initial K=11 J and final K=8/3 J, a decrease of 25/3 J into internal forms. Rotor 1 changes L from 6 to 8/3, while rotor 2 changes from −2 to 4/3: the changes are −10/3 and +10/3. The whole-system L is conserved but the L of each rotor is not. This original collision comparison is theoretical only.

Worked original case

With initial omega=2 and final radius 1 m, Ii=9, If=3 and L=18. Final omega=6 rad/s. K rises from 18 to 54 J, requiring 36 J of internal work under the endpoint assumptions.

Investigation

Compare final radii 2,1,.5 m at initial omega=2. Derive L, final speed and energy change before checking. Then audit the two-rotor coupling example with separately selected subsystems and show their angular impulses cancel. Identify where assuming conserved kinetic energy would contradict the stated interaction.

Public numerical transfer practice

Try a new practice case

A system has initial I=6 kg m² and omega=2 rad/s. With no external torque its inertia becomes 3 kg m². What is final angular velocity in rad/s?

Numerical result in the requested units Check this practice result

This public, unsaved check is not a secure assessment. Explain the derivative, integral or boundary in the retained course evidence workspace.

Sources and evidence boundary

Original derivations and assigned numerical cases checked 30 October 2026 against the official AP Physics C Mechanics framework and OpenStax angular momentum. These are supplied or generated records, not physical measurements or performed laboratory time. Do not construct rotating equipment, release loads or conduct launch experiments. Public feedback and retained self-reports do not certify independent mastery.