Rotational inertia: where the mass matters
You will be able to: Calculate inertia for point-mass systems and compare mass distributions.
Why is a mass harder to spin when it is farther from the axis?
Two identical small masses attach to a light rod on either side of its center. Move each mass twice as far from the axle. The total mass has not changed, but the rotational inertia becomes four times larger.
A useful starting point: Choosing a system →
Words and symbols before equations
- Rotational inertia I
- Resistance to angular acceleration about a specified axis; unit kg·m².
- Point-mass model
- An object whose own size is negligible for the calculation.
- Perpendicular distance r
- Distance to the rotation axis, not distance along that axis.
- Σmr²
- Add mass times squared distance for every point mass.
What this picture assumes
Two 1 kg point masses at ±r on a negligible-mass rigid rod; axis perpendicular to the drawing through the center. Each setting is a separate fixed geometry, not a model of masses sliding during rotation.
Connect the picture to the physics
For one point mass, I=mr². Doubling mass doubles its contribution; doubling distance multiplies it by four. Distance is squared, so opposite sides of an axis both make positive contributions.
For several particles, I=Σmr². Include each mass once and use its distance to the same axis. We use five or fewer point objects in a plane. A mass on the axis contributes zero in the point-mass approximation.
An extended body is not generally a point mass at its center. A hoop has more mass far from the center than a solid disk of the same mass and radius, so its central-axis inertia is larger. When an extended-body formula is needed here, it is supplied; memorizing a catalog is not the aim.
A worked example, step by step
Two 1 kg point masses are each 0.5 m from a central axis. Find total I. Then move both to 1 m from the same axis.
- Each original contribution is mr²=1(0.5²)=0.25 kg·m².
- Total I=0.25+0.25=0.5 kg·m².
- At 1 m, each contributes 1 kg·m², giving total I=2 kg·m².
- Doubling distance multiplied I by four while total mass remained 2 kg.
Rotational inertia is not mass alone. State the axis and square each distance before adding.
Can two objects with the same mass and outer radius have different inertia?
Compare with an explanation
Yes. Their mass can be distributed differently relative to the axis, as with a hoop and disk.
Predict. Change one thing. Explain.
Move both 1 kg point masses outward together. Compare r=0.2, 0.4 and 0.8 m. Predict the factors of change in I before reading the graph.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Each point contributes 0.25 kg·m²; total I=0.5 kg·m². The rod is modeled with negligible mass.
Two 1 kg point masses at ±r on a negligible-mass rigid rod; axis perpendicular to the drawing through the center. Each setting is a separate fixed geometry, not a model of masses sliding during rotation.
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 questionA 1 kg point mass is 0.2 m from an axis and a 2 kg point mass is 0.4 m away. (a) Find each contribution. (b) Find total I. (c) Find total I if both distances double. (d) Explain why contributions cannot cancel across opposite sides.
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Compare with the answer and four-point rubric
- 1 point: 0.04 and 0.32 kg·m².
- 1 point: 0.36 kg·m².
- 1 point: 1.44 kg·m².
- 1 point: Each term uses a squared distance and positive mass, so both contributions are nonnegative.
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 1Inertia units?
kg·m².
RECALL 2Which distance enters mr²?
Perpendicular distance to the axis.
RECALL 3Same mass guarantees same I?
No; distribution and axis matter.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Rotational inertia: where the mass matters
- Point mass: I=mr². Collection: I=Σmr².
- I depends on mass distribution and the chosen axis.
Remember: Rotational inertia is not mass alone. State the axis and square each distance before adding.
Conditions: Two 1 kg point masses at ±r on a negligible-mass rigid rod; axis perpendicular to the drawing through the center. Each setting is a separate fixed geometry, not a model of masses sliding during rotation.
Refresh Kid · Unit 5 · Objectives 5.4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.4, objectives 5.4.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.
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