Energy in a spinning object
You will be able to: Calculate rotational kinetic energy and explain its dependence on inertia and angular speed.
Can a wheel have kinetic energy while its center stays still?
A bicycle wheel spins on a stand. Its center stays in one place, but every part away from the axle moves. It therefore stores kinetic energy. Spinning it twice as fast stores four times as much energy, even though the wheel itself has not changed.
A useful starting point: Rotational inertia →
Words and symbols before equations
- Rotational inertia I
- How mass is distributed about a specified axis; measured in kg·m².
- Angular velocity ω
- Signed turning rate in rad/s; angular speed is its magnitude.
- Rotational kinetic energy K_rot
- Energy associated with rotation, in joules (J).
- Scalar
- A quantity without a direction; kinetic energy is nonnegative.
What this picture assumes
Rigid body rotating about a fixed axis. Each setting is a separate state; changes of inertia are not modeled as a physical process. K is nonnegative for either turning sense.
Connect the picture to the physics
Imagine dividing the wheel into small masses. Each small mass has kinetic energy ½mv². A mass at radius r has speed rω, so its energy is ½mr²ω². Adding all those contributions gives K_rot=½Iω² because I is the sum of the mr² contributions.
At fixed I, energy depends on the square of angular speed. Reversing the turning direction leaves ω² unchanged. Doubling I at the same ω doubles energy; doubling ω at the same I quadruples it. These comparisons specify what stays fixed.
The choice of axis matters for I. Use I and ω for the same rotation. Rotational kinetic energy is not torque: torque describes a turning interaction, whereas energy describes a state and transfers through work. A stationary center of mass does not mean every part is stationary.
A worked example, step by step
A wheel has I=2 kg·m². Find its rotational kinetic energy at +3 rad/s, then at −6 rad/s.
- Identify fixed-axis rotation and use K=½Iω².
- Initially K=½(2)(3²)=9 J.
- Finally K=½(2)(−6)²=36 J.
- The energy is four times larger because speed doubled. The reversed direction does not make the energy negative.
Kinetic energy uses ω². A clockwise rotation does not carry negative kinetic energy.
A spinning disk has zero center-of-mass velocity. Must its kinetic energy be zero?
Compare with an explanation
No. Its parts move around the center, so it can have rotational kinetic energy.
Predict. Change one thing. Explain.
Hold I fixed. Compare +3, −3 and +6 rad/s. Predict the energies before changing the slider. Then hold ω fixed and double I; explain why that comparison gives a different factor.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
I=2 kg·m²; ω=3 rad/s; K=9 J. Doubling this angular velocity at fixed I gives 36 J.
Rigid body rotating about a fixed axis. Each setting is a separate state; changes of inertia are not modeled as a physical process. K is nonnegative for either turning sense.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant physical relationship or contact condition 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 questionTwo identical wheels each have I=0.5 kg·m². A turns at +4 rad/s and B at −8 rad/s. (a) Calculate K_A. (b) Calculate K_B. (c) Compare them by a ratio. (d) Explain why the sign of ω does not determine the sign of energy.
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Compare with the answer and four-point rubric
- 1 point: K_A=½(0.5)(16)=4 J.
- 1 point: K_B=½(0.5)(64)=16 J.
- 1 point: K_B/K_A=4, consistent with double the speed.
- 1 point: Energy is a scalar sum of ½mv² contributions; ω is squared.
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 1Rotational kinetic energy formula?
K_rot=½Iω².
RECALL 2Can K_rot be negative?
No; I≥0 and ω²≥0.
RECALL 3What stays fixed when doubling speed quadruples energy?
The rotational inertia about the same axis.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Energy in a spinning object
- K_rot=½Iω² (J), with I and ω about the same axis.
- At fixed I: doubling |ω| multiplies K by four.
Remember: Kinetic energy uses ω². A clockwise rotation does not carry negative kinetic energy.
Conditions: Rigid body rotating about a fixed axis. Each setting is a separate state; changes of inertia are not modeled as a physical process. K is nonnegative for either turning sense.
Refresh Kid · Unit 6 · Objectives 6.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 6.1, objectives 6.1.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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