Infer angular change from net torque and inertia

1. Frame the question

For a rigid body rotating about a fixed inertial axis with constant inertia, the angular form of Newton’s second law is τnet=Iα. It relates the net external turning effect to angular acceleration, not directly to angular velocity. Use the same axis when computing inertia and every torque; mismatched axes describe different problems.

2. Build the mechanism

For one point particle constrained at radius r, the tangential law is Fₜ=maₜ=mrα. Multiplying by r gives rFₜ=mr²α. Adding the particles’ equations for a rigid system yields the sum of external torques equal to (Σmr²)α; internal interactions cancel in the system torque balance. Radial forces do not provide torque about that axis, though they can be required to maintain circular paths.

If net torque is positive while angular velocity is negative, the magnitude of the angular velocity initially decreases. If the torque persists, the rotor can stop and reverse. Calling the torque “a force in the direction of motion” would obscure this sign distinction. Similarly, zero net torque preserves an already nonzero angular velocity instead of requiring rest.

Increasing inertia at the same torque reduces angular acceleration. This comparison assumes the distribution remains rigid during each run. A person changing body configuration while spinning is a different problem involving changing inertia and angular momentum, treated in the next unit. A real axle’s friction would belong in the net torque rather than being ignored by the equation.

3. Work an example

Take I=2 kg m², τnet=+4 N m and initial ω=−2 rad/s. Then α=+2 rad/s². The rotor stops at 1 s, finishes 3 s at +4 rad/s, and has signed angular displacement −2×3+½×2×9=3 rad. Its early clockwise motion is entirely consistent with counterclockwise acceleration.

4. Predict and investigate

Predict first: Does positive net torque guarantee positive angular velocity at every time?

Compare net torques 0 and +4 N m with inertia 2 kg m². Sketch angular velocity first, identify the moment of reversal, and compare signed graph area. Next double inertia at the same torque and predict how the reversal time changes. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

A constant net torque −6 N m acts on inertia 3 kg m². Find angular acceleration.

Your numerical result (rad/s²) 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

Which measurements would allow you to distinguish a larger inertia from an opposing friction torque if both reduce angular acceleration?

Retain your first response, an extended-body diagram or labeled graph, units, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic model captures are not a performed physical experiment.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 5 topic mapping: 5.6. Checked 30 October 2026. Planar clockwise/counterclockwise analysis; supplied formulas for extended bodies, no required calculus or three-dimensional torque-vector direction. Framework cross-reference, not AP authorization.