Connect a shared rotation to different point motions

1. Frame the question

Choose two marks on a rigid rotating surface at different distances from a fixed axis. Each mark completes a turn in the same time, so both have the same angular velocity. Their arc lengths differ: s=rθ for angle in radians. Dividing by time gives tangential speed v=r|ω|. A larger radius gives more distance per turn, not more turns per second.

2. Build the mechanism

A changing angular velocity produces tangential acceleration aₜ=rα. Even when angular speed is constant, the point velocity changes direction, producing inward centripetal acceleration aᶜ=v²/r=rω². These describe perpendicular components. The magnitude of the full acceleration is the square root of their squared magnitudes, not their ordinary sum.

At a point on the right of the axis, positive counterclockwise velocity is upward. Positive angular acceleration gives an upward tangential component there, while the centripetal component points left toward the axis. At a point on the left, the corresponding directions change even though ω and α are shared. This is why “all points have the same velocity” is false for a rotating rigid body.

The formulas refer to motion relative to a fixed axis. If the axis itself translates, add its velocity vector to each point’s rotational velocity in the chosen frame. Rolling bodies will require that additional bookkeeping later. A point on the fixed axis has zero rotational linear speed; the body can still have nonzero angular velocity.

3. Work an example

For r=0.4 m, ω=3 rad/s and α=2 rad/s², the outer point has speed 1.2 m/s, tangential acceleration 0.8 m/s² and inward acceleration 3.6 m/s². The point at r/2 has half each linear quantity. The outer total acceleration is √(0.8²+3.6²)≈3.69 m/s², not 4.4 m/s².

4. Predict and investigate

Predict first: If ω doubles at fixed α and radius, do tangential and centripetal acceleration change by the same factor?

Hold radius at 0.4 m and compare ω=2 with ω=4 rad/s. Draw the two perpendicular acceleration components at the rightmost point and compare their magnitudes. Then halve radius while retaining the angular quantities; distinguish an angular comparison from a linear one. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

At radius 0.3 m and angular speed 4 rad/s, find inward acceleration.

Your numerical result (m/s²) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

Why can a point have zero instantaneous speed yet nonzero tangential acceleration? What is its centripetal acceleration at that instant?

Retain your first response, an extended-body diagram or labeled graph, units, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic model captures are not a performed physical experiment.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 5 topic mapping: 5.2. Checked 30 October 2026. Planar clockwise/counterclockwise analysis; supplied formulas for extended bodies, no required calculus or three-dimensional torque-vector direction. Framework cross-reference, not AP authorization.