Measure inertia from torque and acceleration
You will be able to: Design a torque–acceleration investigation and interpret slope, intercept and uncertainty.
How could a graph reveal a wheel’s rotational inertia?
Apply several known torques to the same wheel and measure how quickly its angular velocity changes. A graph of applied torque against angular acceleration can reveal inertia—and can expose a resistive torque that a single measurement might hide.
A useful starting point: A pulley connects translation and rotation →
Words and symbols before equations
- Independent variable
- The quantity deliberately varied, here applied torque.
- Angular acceleration measurement
- Slope of a measured ω–t segment.
- Graph slope
- Change in vertical coordinate divided by change in horizontal coordinate.
- Intercept
- Vertical-axis value when the horizontal coordinate is zero.
- Uncertainty
- A justified estimate of measurement limitations, not an arbitrary percentage.
What this picture assumes
Synthetic positive-rotation trials. Applied torques 3, 5 and 7 N·m; constant opposing friction torque. Graph is applied torque versus angular acceleration. It is not a dataset from real students.
Connect the picture to the physics
Keep the same wheel, axis and mass distribution. Apply a measured tangential force F at known radius r, so τ_applied=rF. Determine α from angular-velocity measurements over short intervals. Repeat for several torque values and repeated trials.
For positive rotation with approximately constant opposing friction τ_f, τ_applied−τ_f=Iα. Plot applied torque vertically against α horizontally: slope is I and vertical intercept is τ_f. If plotting α vertically against net torque, the slope would instead be 1/I.
Estimate uncertainties in force, radius, angle and timing, and inspect whether a straight-line model is justified. Curvature or inconsistent intercepts may indicate changing friction, slipping, measurement bias or changing geometry. Synthetic values below demonstrate the method; they are not measured proof of a real apparatus’s behavior.
A worked example, step by step
Synthetic trials give (α,τ_applied)=(1,3), (2,5), (3,7), with α in rad/s² and torque in N·m. Determine I and the assumed constant opposing torque.
- Slope using two separated points: I=(7−3)/(3−1)=2 kg·m².
- Intercept: τ_f=τ_applied−Iα=3−2(1)=1 N·m.
- The middle point also fits 5=2(2)+1.
- For real trials, repeat measurements and estimate uncertainty before claiming agreement. Using τ_applied/α from one point would incorrectly give 3 kg·m² because it ignores friction.
Read the axis labels before interpreting slope. Applied torque divided by α is not I when an unaccounted resistive torque is present.
If you plot α vertically and net torque horizontally, is the slope I?
Compare with an explanation
No. Since α=τ_net/I, that graph’s slope is 1/I.
Predict. Change one thing. Explain.
Change the synthetic wheel’s true inertia and opposing friction. Observe how slope and intercept change separately. State what would have to be measured in an actual experiment.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Model slope I=2 kg·m²; intercept=1 N·m. Synthetic (α,τ) pairs: (1, 3); (2, 5); (3, 7). The line to α=0 is a model extrapolation.
Synthetic positive-rotation trials. Applied torques 3, 5 and 7 N·m; constant opposing friction torque. Graph is applied torque versus angular acceleration. It is not a dataset from real students.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use an angular-motion or torque relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionData pairs (α,τ_applied) are (1,4), (2,7), (3,10). (a) Find slope. (b) Find intercept. (c) Interpret both. (d) Name one measurement uncertainty and one controlled condition.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Slope=(10−4)/(3−1)=3 kg·m².
- 1 point: Intercept=4−3(1)=1 N·m.
- 1 point: I=3 kg·m² and assumed opposing torque=1 N·m.
- 1 point: For example, timing/force uncertainty; keep the rotation axis and mass distribution fixed.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Slope of applied torque versus α?
I, under the stated constant-friction model.
RECALL 2Why take several trials?
To evaluate the relationship and uncertainty, rather than trust a single ratio.
RECALL 3What must remain fixed?
The axis and mass distribution, hence I, within one experiment.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Measure inertia from torque and acceleration
- τ_applied=Iα+τ_f for a constant opposing torque during positive rotation.
- Slope of τ_applied versus α is I; intercept is τ_f.
Remember: Read the axis labels before interpreting slope. Applied torque divided by α is not I when an unaccounted resistive torque is present.
Conditions: Synthetic positive-rotation trials. Applied torques 3, 5 and 7 N·m; constant opposing friction torque. Graph is applied torque versus angular acceleration. It is not a dataset from real students.
Refresh Kid · Unit 5 · Objectives 5.6.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.6, objectives 5.6.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Fall-2026 corrections also checked. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
Want to work through this with a tutor?
Bring your question about measure inertia from torque and acceleration. Your explanation and answers remain free to access.