Newton’s second law for rotation
You will be able to: Use net torque and rotational inertia to determine angular acceleration.
What sets how quickly a wheel’s rotation changes?
The same turning push changes a light wheel’s rotation more quickly than a wheel with a large rotational inertia. Conversely, doubling the net turning effect on one unchanged wheel doubles its angular acceleration.
A useful starting point: Rotational inertia →
Words and symbols before equations
- Net torque Στ
- Combined signed turning effect of all external forces about the chosen axis.
- Rotational inertia I
- Inertia about that same axis, in kg·m².
- Angular acceleration α
- Rate of change of signed angular velocity, in rad/s².
What this picture assumes
Rigid body about a fixed axis, fixed rotational inertia for each trial. ω₀=+2 rad/s. Constant net torque includes all applied and resistive contributions; CCW positive.
Connect the picture to the physics
For a rigid body about a fixed axis with constant I, Στ=Iα. Rearrange to α=Στ/I. Include resistive torques before dividing by I. Torque and inertia must refer to the same axis.
At fixed I, doubling net torque doubles α. At fixed net torque, doubling I halves α. The turning sign of α follows net torque; whether speed increases depends on the existing sign of ω.
If net torque and I are constant, α is constant and the Unit 5 kinematics equations apply. This relation does not replace a translational force balance when the center of mass also accelerates. We analyze one plane and avoid changing-inertia spin dynamics in this unit.
A worked example, step by step
A wheel has I=2 kg·m², motor torque +6 N·m and friction torque −2 N·m. Initially ω=+2 rad/s. Find α and ω after 3 s if the torques stay constant.
- Net torque is 6−2=+4 N·m.
- α=Στ/I=4/2=+2 rad/s².
- ω=ω₀+αt=2+2(3)=+8 rad/s.
- The wheel speeds up counterclockwise. Using the motor torque alone would overestimate α because it omits friction.
Use net torque, not just the largest or most visible applied torque.
Can negative net torque act on a wheel rotating counterclockwise?
Compare with an explanation
Yes. It gives negative α and initially slows a positive ω; if maintained, it may reverse the wheel.
Predict. Change one thing. Explain.
Compare net torques +4, 0 and −4 N·m at I=2 kg·m². Then double I at fixed torque. The graph begins at ω₀=+2 rad/s; explain any slowdown and reversal.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Net torque=4 N·m; I=2 kg·m²; α=2 rad/s². From ω₀=2 rad/s, ω(4 s)=10 rad/s.
Rigid body about a fixed axis, fixed rotational inertia for each trial. ω₀=+2 rad/s. Constant net torque includes all applied and resistive contributions; CCW positive.
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 wheel has I=0.5 kg·m², applied torque +3 N·m and opposing torque −1 N·m. Initially it rests. (a) Find net torque. (b) Find α. (c) Find ω after 2 s. (d) Find Δθ in those 2 s.
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Compare with the answer and four-point rubric
- 1 point: +2 N·m.
- 1 point: α=2/0.5=+4 rad/s².
- 1 point: ω=+8 rad/s.
- 1 point: Δθ=½(4)(2²)=+8 rad, assuming constant torques and inertia.
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 second law?
Στ=Iα.
RECALL 2Double I at fixed net torque?
α is halved.
RECALL 3Which axis is used for I?
The same axis used to calculate the torques.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Newton’s second law for rotation
- Στ=Iα about a fixed axis for constant I.
- α=Στ/I; if α is constant, ω=ω₀+αt.
Remember: Use net torque, not just the largest or most visible applied torque.
Conditions: Rigid body about a fixed axis, fixed rotational inertia for each trial. ω₀=+2 rad/s. Constant net torque includes all applied and resistive contributions; CCW positive.
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.
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