Infer rotational inertia from a coupled motion model

1. Frame the question

A hanging mass and a fixed-axis pulley require separate system diagrams. For the descending mass, weight mg acts down and tension T up, giving mg−T=ma. For the pulley, string tension acts tangentially at radius R, giving TR=Iα when bearing friction is neglected. The axle forces do not contribute torque about its own fixed axis.

2. Build the mechanism

For a taut inextensible string that does not slip, linear acceleration of the mass equals tangential rim acceleration: a=Rα. Substitute α=a/R and T=Ia/R² into the mass equation to get a=mg/(m+I/R²). The term I/R² has mass units, which is a useful dimensional check. Tension is below weight because the mass accelerates; setting T=mg would contradict that motion.

Rearranging the same relation gives I=mR²(g/a−1). This is an inference conditional on the model, not a direct measurement of inertia. If friction opposes rotation, a smaller observed acceleration can be misinterpreted as larger inertia. An incorrect effective string radius or slipping string can also bias the conclusion.

Use only virtual or supplied observations here; do not suspend weights or construct a spinning apparatus. A genuine practical study requires an approved risk assessment, supervision and calibrated measurements. To improve a supplied-data inference, compare several masses, retain the raw acceleration estimates and uncertainty, and test whether one inertia explains them all. Agreement is supporting evidence under assumptions, not proof that all neglected effects vanish.

3. Work an example

With m=1 kg, R=0.10 m, g=10 m/s² and I=0.02 kg m², acceleration is 10/(1+2)=10/3 m/s². Tension is 10−10/3=20/3 N and α=100/3 rad/s². Then TR=2/3 N m equals Iα. If a supplied acceleration lies between 3.2 and 3.4 m/s² with exact m,R,g, inferred I lies between 0.01941 and 0.02125 kg m²; the inverse relation reverses the endpoint order.

4. Predict and investigate

Predict first: At fixed pulley inertia and radius, will increasing hanging mass make acceleration approach g or move farther from it?

Use at least three hanging masses while holding inertia fixed. Predict acceleration and tension, verify both Newton-law equations, and recover inertia from each synthetic acceleration. Then explain how a fixed opposing axle torque would alter the equation to a=(mg−τf/R)/(m+I/R²), and why treating it as zero could create mass-dependent inferred inertias. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

A 2 kg hanging mass has measured acceleration 5 m/s², with R=0.10 m and g=10 m/s² under the ideal assumptions. Infer pulley inertia.

Your numerical result (kg m²) 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 independent observation could test the no-slip condition? Why would repeated perfectly matching values generated by this same model not establish independent apparatus validity?

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: 5.2; 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.