Connect torque-time evidence to angular-momentum change

1. Frame the question

External torque transfers angular momentum to or from a chosen system. For a constant torque, angular impulse is τΔt. For a variable pulse it is the signed torque-time area. The angular impulse equals ΔL, with units N m s, equivalent to kg m²/s. This equality is different from the torque-angle work relation.

2. Build the mechanism

The slope of an angular-momentum-time graph gives net external torque: ΔL/Δt is its interval average, and the local slope gives the instantaneous value. A flat L graph can represent steady nonzero spin; it means no net angular-momentum transfer, not necessarily zero angular momentum. Rising and falling segments identify opposite signed torques under the chosen planar convention.

For constant inertia, ΔL=IΔω, so a pulse predicts final angular velocity. If the mass distribution changes, use endpoint values Iinitialωinitial and Ifinalωfinal rather than pretending one I multiplies Δω. Internal torques may change an individual part’s angular momentum even when they cancel for the combined system.

A conservation test needs both a system boundary and an uncertainty statement. If observed angular momentum changes by 0.06 kg m²/s while the bounded error in each before/after value is 0.04, zero change is compatible with a worst-case difference bound 0.08. That is not proof of exact isolation. A resolved discrepancy would motivate examination of external torque and the measurement model, not automatic dismissal of the law.

3. Work an example

A triangular external torque pulse peaks at +4 N m and lasts 1 s. Its angular impulse is ½×4×1=2 N m s. A rotor with I=2 kg m² starts at +1 rad/s, so initial L=2 and final L=4 kg m²/s, giving final ω=2 rad/s. A rectangular pulse with the same peak and duration would deliver twice the impulse.

4. Predict and investigate

Predict first: If peak torque doubles and duration halves, which quantity stays fixed: angular impulse, peak angular acceleration or elapsed time?

Compare pulses (4 N m,1 s) and (2 N m,2 s), then reverse their signs. Reconstruct final angular momentum before checking. Sketch a separate L-time line rising from −2 to +2 over 2 s and then flat: its first average torque is +2 N m and its later torque is zero. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

A fixed-inertia rotor starts with L=−3 kg m²/s and receives +5 N m s angular impulse. Find final L.

Your numerical result (kg m²/s) 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

What would you measure to distinguish a small external angular impulse from an error in the inferred inertia?

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: 6.3; 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.