Conservation of Angular Momentum

A point mass m at radius r has moment of inertia I = mr². With no external torque about the fixed axis, its signed angular momentum L = Iω stays constant. A radial actuator can change rotational kinetic energy without exerting torque about that axis. Energy conservation for the rotating mass alone is therefore not assumed.

The initial and final radii are prescribed configurations. I₀ = mr₀², I_f = mr², ω_f = L/I_f and K = L²/(2I). External radial work equals K_f − K₀. This is the change in rotational energy; radial kinetic energy and the actuator are excluded.

Worked example: inward change

For m = 1 kg, initial radius 2 m, final radius 1 m and initial angular speed 1 rad/s, the prescribed inward change increases the angular speed. Each value follows directly from inertia and conserved angular momentum.

Initial inertia is 1 × 2² = 4 kg·m²; final inertia is 1 × 1² = 1 kg·m².
Angular momentum is 4 × 1 = 4 kg·m²/s, so final angular speed is 4/1 = 4 rad/s.
Initial energy is 4 × 1²/2 = 2 J; final energy is 1 × 4²/2 = 8 J.
The external radial work is 8 − 2 = 6 J. Conserved angular momentum does not imply conserved kinetic energy.

Native experiment

Open Conservation of Angular Momentum. Use these exact labelled raw controls; results update on each valid edit, while invalid inputs replace the calculation with a visible error.

Set Mass m = 1, Initial radius r0 = 2, Final radius r = 1 and Initial angular speed omega0 = 1. Check L = 4, final angular speed 4 and work 6 J.
Set Mass m = 2, Initial radius r0 = 1, Final radius r = 2 and Initial angular speed omega0 = 3. Read inertias 2 and 8, angular momentum 6, final speed 0.75 rad/s, energies 9 and 2.25 J and work −6.75 J.
Set Mass m = 3, Initial radius r0 = 1, Final radius r = 1 and Initial angular speed omega0 = -2. Read L = −6, final angular speed −2 and both energies 6 J; clockwise direction is retained.
Set Initial angular speed omega0 = 0. Angular momentum, both energies, final angular speed and work are all 0; no rotation arrows are shown.
Set Initial radius r0 and Final radius r both to 1.5, Mass m = 2 and Initial angular speed omega0 = -4. Final angular speed equals −4, both energies are 36 J and work is 0.
Set Final radius r = 0. The actual domain error replaces results; no infinite spin or hidden result is supplied. Restore r = 0.5, set r0 = 3, m = 5 and omega0 = 4. Read final angular speed 144 rad/s, final energy 12960 J and work 12600 J.
Pan the diagram and choose Reset view. The four inputs remain unchanged. Clear Mass m, inspect the error, then restore 1 to recover the comparison.

Model limitations

The module compares two static point-mass configurations around one fixed axis, not a deformable human body, a full flywheel or a time-dependent contraction. The diagram uses equal spatial scales; its fixed illustrative angles and direction-only arc arrows are not elapsed-time positions or a numerical integration. Radii are positive and bounded, so a zero-radius singularity is rejected. Displayed decimals use eleven significant digits and ordinary floating-point arithmetic.

References

The torque-free conservation law, changing moment of inertia and resulting changes in rotation are discussed in OpenStax University Physics Volume 1, Section 11.3. The bounded point-mass comparison here is a deterministic application of that law.

Conservation of Angular Momentum