Balanced torque can mean steady rotation
You will be able to: Distinguish constant angular velocity from rest and from translational equilibrium.
Does zero net torque mean a wheel must stop?
A wheel on an ideal frictionless axle can keep spinning without a net torque. Balanced turning effects mean its angular velocity does not change; they do not require that angular velocity to be zero.
A useful starting point: Adding torques about one axis →
Words and symbols before equations
- Rotational equilibrium
- Zero net torque, giving constant angular velocity for the fixed-inertia system here.
- Translational equilibrium
- Zero net force, giving constant center-of-mass velocity.
- Static equilibrium
- A system at rest that satisfies both force and torque balance.
What this picture assumes
Fixed inertia and zero net torque about a fixed axis. The graph illustrates constant angular velocity; it is not a complete translational force model.
Connect the picture to the physics
Choose a fixed axis and a rigid system whose inertia is constant. If Στ=0, then α=0 and ω stays constant. That constant may be positive, negative or zero. For a real wheel with axle friction, continued steady rotation requires a driving torque that balances friction.
Force balance and torque balance are independent tests. A pair of equal and opposite forces at different locations can have zero net force but nonzero torque. A force through the center of mass can accelerate that center without producing torque about it.
Static equilibrium adds the initial condition of rest to both balances. A beam at rest needs ΣF=0 and Στ=0. Checking only torques misses whether the entire beam accelerates upward or downward.
| Question | Translational equilibrium | Rotational equilibrium |
|---|---|---|
| Balance required | ΣF=0 | Στ=0 about the relevant axis |
| Consequence | Constant CM velocity | Constant angular velocity for fixed I |
| Must the system be at rest? | No | No |
A worked example, step by step
A wheel rotates at +3 rad/s while a motor supplies +2 N·m and friction supplies −2 N·m. Describe its angular motion. What if the motor is removed?
- Net torque with the motor is 2−2=0 N·m.
- For fixed I, α=0, so ω remains +3 rad/s; the wheel need not stop.
- Without the motor, net torque is −2 N·m while friction opposes its positive motion.
- The wheel’s positive angular speed decreases. The zero-net-torque description no longer applies.
Zero net torque means no change in angular velocity, not necessarily zero angular velocity.
Can a body be in translational equilibrium but not rotational equilibrium?
Compare with an explanation
Yes. Equal and opposite forces at different locations can cancel as forces while producing a net turning effect.
Predict. Change one thing. Explain.
With net torque held at zero, choose positive, zero or negative angular velocity. Explain why every horizontal ω–t graph still represents rotational equilibrium.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Net torque=0; α=0; ω remains 2 rad/s. Steady counterclockwise rotation. Net force must be checked separately.
Fixed inertia and zero net torque about a fixed axis. The graph illustrates constant angular velocity; it is not a complete translational force model.
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 fixed-axis wheel has motor torque +5 N·m and resistive torque −5 N·m. Initially ω=+2 rad/s. (a) Find net torque. (b) Find α. (c) Find ω after 3 s. (d) Explain whether torque balance proves translational equilibrium for an arbitrary body.
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Compare with the answer and four-point rubric
- 1 point: 0 N·m.
- 1 point: 0 rad/s² for fixed nonzero I.
- 1 point: +2 rad/s.
- 1 point: No. Net force must be checked separately.
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 1Does rotational equilibrium require rest?
No; constant ω is sufficient for fixed I.
RECALL 2Does zero net force ensure zero net torque?
No; application points matter.
RECALL 3What extra condition defines static equilibrium?
The system is initially at rest and both balances hold.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Balanced torque can mean steady rotation
- Rotational equilibrium: Στ=0, α=0 for fixed I.
- Static equilibrium requires ΣF=0 and Στ=0, with initial rest.
Remember: Zero net torque means no change in angular velocity, not necessarily zero angular velocity.
Conditions: Fixed inertia and zero net torque about a fixed axis. The graph illustrates constant angular velocity; it is not a complete translational force model.
Refresh Kid · Unit 5 · Objectives 5.5.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.5, objectives 5.5.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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