Angular momentum conservation
Angular momentum is the rotational quantity conserved when external torque is zero. For a fixed axis, it is the product of inertia and angular velocity.
Think like a programmer
Model checklist
- Inputs
- Moment of inertia, angular velocity, and external torque.
- State
- Angular momentum.
- Rule
- Update momentum by applied torque times time.
- Output
- Rotation rate after a shape or torque change.
- Check
- With no external torque, total angular momentum remains constant.
Torque updates angular motion
Change torque and moment of inertia. The same torque produces less angular acceleration for a harder-to-rotate body.
Torque 6 N·m; moment of inertia 2 kg·m²; angular acceleration 3.00 rad/s²; rotational energy at 3 rad/s 9.00 J.
Make the system boundary do the work
For a fixed-axis model, record scalar angular momentum (L=I\omega), applied external torque, and the time interval:
\[L_{n+1}=L_n+\tau_{\mathrm{ext},n}\Delta t, \qquad r_L=L_n-L_0-\sum_{k<n}\tau_{\mathrm{ext},k}\Delta t_k.\]The residual (r_L) is a test output. It is zero in an ideal isolated fixture within a scale-aware numerical tolerance. A torque between two components is internal only if both components are inside the chosen boundary. A motor, a bearing attached to Earth, or a hand can transfer angular momentum across the boundary even if the visible rotor is the only object on screen.
If inertia changes while external torque is zero, solve (I_0\omega_0=I_1\omega_1) rather than holding both (L) and (\omega) constant. The kinetic energy can change because internal work rearranges the mass distribution; log that transfer or state the actuator/reservoir that provides it.
Prefer fixtures to visual steadiness
Test zero external torque, a known constant-torque ramp, doubled inertia at equal initial momentum, and equal/opposite torques in a two-part isolated rotor model. Compare fixed physical time at smaller steps, inspect the accumulated torque impulse and residual, and preserve units and sign conventions in the output packet.
In three dimensions, (\mathbf L=\mathbf I\boldsymbol\omega), and neither vectors nor torque need be aligned. A scalar spinner cannot establish general rigid-body conservation. It is a constrained model whose axis, supports, and excluded angular-momentum channels must be declared.
Try this experiment
Prediction: If inertia falls with no external torque, angular speed rises.
Hold the torque at zero. Calculate the angular speed needed to preserve angular momentum when the inertia is halved.Where this model breaks
Summary
Treat angular momentum as state, integrate only declared external torque, and inspect the impulse residual, boundary, changing-inertia transfer, analytic fixtures, and refinement rather than trusting a visually steady rotor.
Glossary
- Angular momentum: rotational momentum.
- External torque: torque from outside the modeled system.
- Principal axis: direction where the inertia relation is especially simple.
Self-check
- What is conserved when external torque is zero?
- Why can a skater spin faster when pulling inward?
- When does a scalar inertia fail?
Sources
- OpenStax, University Physics Volume 1, angular momentum.
Model contract
Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.
\[\text{observable output} = f(\text{inputs},\,\text{state})\]- Inputs
- Quantities you set or measure, with units and useful bounds.
- State
- Values the program must retain to reproduce the next result.
- Rule
- The relationship or update that turns inputs and state into a result.
- Check
- A known limit, unit check, invariant, or measured result that can expose a bad model.
Implement the idea as a model
For Angular Momentum Conservation, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Rolling, Angular Momentum, and Gyroscopes, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.
Guided experiment
Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.
Where this model breaks
This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.
Summary
Treat Angular Momentum Conservation as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.
Glossary
- Input: a measured value or chosen parameter supplied to a model.
- State: the smallest set of values needed to continue or reproduce a model.
- Validation: comparing an output with a known result, limit, invariant, or measurement.
Self-check
- Which values are inputs, and which values must remain state?
- What observable result would tell you the model is behaving as expected?
- Which assumption would you test first before applying the model to a real system?
Model review: turn Angular Momentum Conservation into a test
Represent angular momentum as state, update it with torque, and test the torque-free invariant.
- Name the inputs and units that the rolling, angular momentum, and gyroscopes model needs.
- Separate the state you must keep from values you can calculate when needed.
- Write one rule that maps the current state and inputs to an observable result.
- Choose a limiting case, unit check, invariant, or known result before trusting an output.
- State one assumption you would change before using this simplified model for a real decision.