Same turn, different linear speeds
You will be able to: Relate arc distance and tangential speed to radius and angular motion.
Why does the outside of a carousel move faster?
Two riders stay on the same rotating platform, one twice as far from the center. They complete each revolution together. The outer rider follows a circle twice as long, so moves twice as fast through space.
A useful starting point: Angular velocity and acceleration →
Words and symbols before equations
- Radius r
- Perpendicular distance from the fixed axis, in m.
- Tangential velocity
- Velocity along the instantaneous tangent to the circular path.
- Angular speed |ω|
- Rate of turning in rad/s, shared by all points of the rigid platform.
- Arc distance s
- Distance along the circular path, in m.
What this picture assumes
Two markers on the same rigid disk at radii 0.2 m and 0.4 m, fixed axis. Both share ω. Arrows at the right-hand markers are tangential velocities; counterclockwise rotation is shown.
Connect the picture to the physics
For the same angular turn, s=rθ when θ is the positive angular travel in radians. Dividing by time gives tangential speed v=r|ω|. A larger radius multiplies the same angular speed into a larger linear speed.
Every point on a rigid system shares its angular velocity and angular acceleration about the fixed axis. Their linear speeds differ because their radii differ. A point on the axis has r=0 and zero translational speed in this fixed-axis model.
At the right-hand edge of a CCW wheel, velocity points upward in the drawing. It is tangent to the path, not toward the center. For a wheel whose center also translates, r|ω| describes motion relative to the center; ground-frame velocities require combining the translation and rotation.
A worked example, step by step
A rigid disk turns at 3 rad/s. Markers are at 0.2 m and 0.4 m. Find their speeds and arc distances in 2 s.
- Both markers have ω=3 rad/s; neither has twice the angular speed.
- Inner speed v=(0.2)(3)=0.6 m/s; outer speed v=(0.4)(3)=1.2 m/s.
- In 2 s the disk turns θ=3(2)=6 rad.
- Arc distances are 0.2(6)=1.2 m and 0.4(6)=2.4 m, matching speed times time.
Equal angular velocity does not imply equal linear velocity. Use radians in v=rω.
Does a marker at the center of a fixed-axis disk travel a circular path of nonzero length?
Compare with an explanation
No. Its radius is zero, so its linear speed is zero even while the disk turns.
Predict. Change one thing. Explain.
Change angular speed while comparing two markers at 0.2 m and 0.4 m. Their velocity arrows use the same scale. Explain both the equal ω and the 2:1 linear-speed ratio.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Both markers share ω=3 rad/s. Inner speed=0.6 m/s; outer speed=1.2 m/s. The outer marker moves twice as fast.
Two markers on the same rigid disk at radii 0.2 m and 0.4 m, fixed axis. Both share ω. Arrows at the right-hand markers are tangential velocities; counterclockwise rotation is shown.
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 questionTwo markers on one disk have radii 0.1 and 0.3 m. The disk turns at 5 rad/s for 2 s. (a) State each ω. (b) Find each speed. (c) Find the angle turned. (d) Find each arc distance.
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Compare with the answer and four-point rubric
- 1 point: Both have ω=5 rad/s.
- 1 point: 0.5 m/s and 1.5 m/s.
- 1 point: 10 rad.
- 1 point: 1 m and 3 m.
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 rate is shared by a rigid disk?
Angular velocity.
RECALL 2Which marker moves faster through space?
The one farther from the fixed axis.
RECALL 3Direction of the instantaneous velocity?
Tangent to the circular path.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Same turn, different linear speeds
- v_t=r|ω|; s=rθ for positive angular travel.
- All points of a rigid body share ω about its fixed axis.
Remember: Equal angular velocity does not imply equal linear velocity. Use radians in v=rω.
Conditions: Two markers on the same rigid disk at radii 0.2 m and 0.4 m, fixed axis. Both share ω. Arrows at the right-hand markers are tangential velocities; counterclockwise rotation is shown.
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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