Solve rotation with constant acceleration
You will be able to: Use rotational kinematics with a stated constant-acceleration assumption.
How far does a spinning wheel turn before stopping?
A wheel starts at +4 rad/s and slows at a steady −1 rad/s². It takes 4 s to reach rest. During that time its average angular velocity is +2 rad/s, so it turns through +8 rad.
A useful starting point: Read rotational graphs: slope and signed area →
Words and symbols before equations
- ω₀
- Initial angular velocity in rad/s.
- Δθ
- Final angle minus initial angle in radians.
- Constant α
- Angular acceleration with the same value throughout the interval.
- Stopping time
- First time ω reaches zero; not necessarily the end of a continued motion.
What this picture assumes
ω₀=+4 rad/s and θ₀=0. The chosen constant acceleration continues through a possible stop and reversal; it is not a brake that automatically disengages at zero speed.
Connect the picture to the physics
Constant acceleration means equal changes in ω in equal time intervals. Then ω=ω₀+αt. The signed area under the straight ω–t line gives Δθ=ω₀t+½αt².
Eliminating time gives ω²=ω₀²+2αΔθ. Use signed α and signed Δθ. Squaring removes the sign of ω, so infer its direction from the physical interval or check with ω=ω₀+αt.
These equations require constant α. They do not describe arbitrary changing torque or inertia. If the acceleration persists past a stop, the wheel reverses; do not keep calling the signed displacement total angular distance.
A worked example, step by step
A wheel starts at +6 rad/s with constant α=−2 rad/s². Find its stopping time and displacement up to the stop.
- At the stop, ω=0. From 0=6−2t, t=3 s.
- Δθ=(6)(3)+½(−2)(3²)=18−9=+9 rad.
- Check with ω²=ω₀²+2αΔθ: 0=36−4Δθ, giving 9 rad.
- The positive displacement fits a wheel that rotates CCW while slowing. Continuing the same α beyond 3 s would produce clockwise motion.
Do not substitute angular speed for signed angular velocity when direction can change.
Can you use the average of initial and final ω for any changing rotation?
Compare with an explanation
No. That average equals the time-average ω for a linear ω–t change, including constant α, but not an arbitrary curve.
Predict. Change one thing. Explain.
Begin with ω₀=+4 rad/s. Set α=−1 and compare t=2, 4 and 6 s. Explain why displacement can decrease after the wheel reverses. Then compare with α=0.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
At 4 s, ω=0 rad/s and Δθ=8 rad for α=-1 rad/s². The signed area gives displacement; it is not total travel after a reversal.
ω₀=+4 rad/s and θ₀=0. The chosen constant acceleration continues through a possible stop and reversal; it is not a brake that automatically disengages at zero speed.
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 starts at +8 rad/s with α=−2 rad/s². (a) Find time to rest. (b) Find Δθ to rest. (c) Find ω at 5 s if α continues. (d) State the assumption allowing these equations.
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Compare with the answer and four-point rubric
- 1 point: 4 s.
- 1 point: Δθ=8(4)−4²=16 rad.
- 1 point: ω=8−2(5)=−2 rad/s, clockwise.
- 1 point: α is constant during each analyzed interval and the axis/view remain 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 1What must be constant?
Angular acceleration.
RECALL 2How do you find the first stopping time?
Set ω=0 in ω=ω₀+αt and select a physically relevant positive time.
RECALL 3Does the squared-velocity equation give direction?
No; check the sign from the motion or another equation.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Solve rotation with constant acceleration
- ω=ω₀+αt; Δθ=ω₀t+½αt².
- ω²=ω₀²+2αΔθ; Δθ=½(ω₀+ω)t.
- All four relations assume constant α.
Remember: Do not substitute angular speed for signed angular velocity when direction can change.
Conditions: ω₀=+4 rad/s and θ₀=0. The chosen constant acceleration continues through a possible stop and reversal; it is not a brake that automatically disengages at zero speed.
Refresh Kid · Unit 5 · Objectives 5.1.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.1, objectives 5.1.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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