AP® Physics 1: Algebra-Based · Unit 5 · Topic 5.2
Connecting Linear and Rotational Motion
Every point on a spinning rigid body shares the same angular velocity, but linear speed grows with distance from the axis: v = omega times r.
Picture a rigid disk turning around a fixed axis. Every point on it sweeps the same angle in the same time, so angular velocity omega is one single number for the whole object. But those points are not all moving at the same actual speed through space. A point far from the center has to trace a bigger circle in that same time, so it covers more distance. That's why the outer edge of a merry-go-round feels like it's flying while the center barely moves: same spin rate, bigger circle.
The link is v = omega times r. Angular velocity tells you radians per second; multiply by the radius and the radians cancel, leaving meters per second, a true linear speed tangent to the circle. Double the radius and you double the speed, even though omega never changed. The same scaling shows up in acceleration: tangential acceleration at = alpha times r, where alpha is angular acceleration. Arc length follows the same logic, s = theta times r, which is really just this same relationship applied to distance instead of rate.
This is the bridge between the rotational variables (theta, omega, alpha) built in earlier rotational kinematics and the linear motion tools from Units 1 and 2. Once you can convert between the two, every linear motion equation has a rotational twin, and quantities like centripetal acceleration (v2/r, which also equals omega2 times r) connect rotation back to the circular motion you already know.
Key terms
- angular velocity (omega)
- The rate an object sweeps angle, in radians per second, the same for every point on a rigid rotating body.
- linear (tangential) speed
- The actual speed of a point along its circular path, v = omega times r, pointing tangent to the circle.
- radius (r)
- The distance from the rotation axis to the point in question, the scaling factor between angular and linear quantities.
- tangential acceleration
- The linear acceleration along the circular path caused by changing angular speed, at = alpha times r.
- rigid body
- An object whose points keep fixed distances from each other, forcing them all to share one angular velocity.
Watch it explained
- How it worksHow does multiplying angular velocity by radius convert a spin into a real speed?
- Worked exampleFind the speed of a point 0.35 m from the axis of a wheel turning at 12 rad/s.
- Common mistakeWhy isn't the angular velocity different for points at different radii on one rigid disk?
- Going furtherWhat happens to the arc length traced by two points at different radii in the same time?
Check yourself
End of topic.