Two jobs of acceleration on a rotating wheel
You will be able to: Separate tangential acceleration from inward centripetal acceleration.
How can a point accelerate when angular speed is constant?
A point on a steadily spinning wheel keeps changing its velocity direction, even when its speed is steady. If the wheel also speeds up, that point has a second acceleration component that changes its speed.
A useful starting point: Same turn, different linear speeds →
Words and symbols before equations
- Tangential acceleration a_t
- Acceleration along the tangent; a_t=rα with a chosen positive tangent.
- Radial acceleration a_c
- Inward acceleration that changes velocity direction; magnitude rω².
- Perpendicular components
- Components at 90° to each other; add as vectors, not simple magnitudes.
What this picture assumes
Marker at the right-hand rim, r=0.5 m. Velocity is instantaneously counterclockwise when ω>0. Radial acceleration points toward the axis; positive tangential acceleration points counterclockwise. These are instantaneous states.
Connect the picture to the physics
Differentiating the rate relationship informally: if every second ω changes by α, the tangential speed changes at r times that rate. Thus the signed tangential acceleration is a_t=rα. For positive α at the right-hand rim, it points upward in our CCW convention.
From circular motion, a_c=v²/r. Substituting v=r|ω| gives a_c=rω², always inward for a moving point. If α=0 but ω≠0, tangential acceleration is zero and radial acceleration remains.
These two perpendicular components have different purposes. Their total magnitude is √(a_t²+a_c²). At an instant with ω=0 and nonzero α, inward acceleration is zero but tangential acceleration need not be.
A worked example, step by step
At r=0.5 m, a wheel has ω=2 rad/s and α=3 rad/s². Find both acceleration components and the total magnitude.
- Tangential: a_t=rα=(0.5)(3)=1.5 m/s² in the CCW tangent sense.
- Radial: a_c=rω²=(0.5)(2²)=2 m/s² inward.
- The components are perpendicular, so a=√(1.5²+2²)=2.5 m/s².
- Adding 1.5+2 would be incorrect because the vectors do not point along the same line.
a=rα describes only the tangential component, not necessarily the total acceleration.
Does α=0 imply zero acceleration for every point on a spinning disk?
Compare with an explanation
No. Points away from the axis still have inward acceleration rω² when ω is nonzero.
Predict. Change one thing. Explain.
First set α=0 and change ω. Then set ω=0 and change α. Finally turn on both and compare the inward and tangent arrows. The picture is an instantaneous state, not a whole trajectory.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
a_c=2 m/s² inward; signed a_t=1 m/s²; total magnitude=2.24 m/s².
Marker at the right-hand rim, r=0.5 m. Velocity is instantaneously counterclockwise when ω>0. Radial acceleration points toward the axis; positive tangential acceleration points counterclockwise. These are instantaneous states.
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 point at r=0.2 m has ω=5 rad/s and α=−10 rad/s². (a) Find a_c. (b) Find signed a_t. (c) Find total magnitude. (d) Describe the directions at the right-hand rim.
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Compare with the answer and four-point rubric
- 1 point: a_c=0.2(25)=5 m/s².
- 1 point: a_t=−2 m/s².
- 1 point: √29≈5.39 m/s².
- 1 point: Radial points left toward the axis; tangent points downward for negative α in the CCW-positive view.
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 1Which component changes speed?
Tangential acceleration.
RECALL 2Which changes velocity direction?
Inward radial acceleration.
RECALL 3Why is radial acceleration nonnegative as a magnitude?
It uses ω²; its direction is separately specified as inward.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Two jobs of acceleration on a rotating wheel
- a_t=rα; a_c=rω²=v²/r.
- a_total=√(a_t²+a_c²) for these perpendicular components.
Remember: a=rα describes only the tangential component, not necessarily the total acceleration.
Conditions: Marker at the right-hand rim, r=0.5 m. Velocity is instantaneously counterclockwise when ω>0. Radial acceleration points toward the axis; positive tangential acceleration points counterclockwise. These are instantaneous states.
Refresh Kid · Unit 5 · Objectives 5.2.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 5.2, objectives 5.2.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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