For a point sitting a distance r from a fixed axis of rotation, the linear distance it actually travels — its arc length — as the system rotates through an angle Δθ is:
Differentiate the arc-length relationship with respect to time, and two of the most useful equations in the whole unit fall right out:
Explore the pulley application directly below, matching this unit's own header art.
The pulley system from this unit's header art. A string wrapped around the pulley's rim connects to a hanging mass — the mass's linear acceleration is exactly the pulley rim's tangential acceleration.
This is exactly how a pulley's rotation gets linked to a hanging mass's linear acceleration — a relationship you'll use constantly starting in Lesson 5.6.
A bicycle wheel of radius 0.35 m rotates at 4 rad/s. Find the linear speed of a point on the wheel's rim.
For a rigid system, every single point shares exactly the same angular velocity and angular acceleration — that's what "rigid" means. But since v = rω and aT = rα both scale directly with r, points farther from the axis move faster and accelerate more, even though they're all spinning at the same rate.
Explore this directly below, matching this unit's header art of masses at different radii on the same rotating disk.
A rigid disk rotating at ω, matching this unit's header art — masses at different radii, all sharing the exact same angular velocity. Watch how differently their linear speeds compare.
Both points complete a revolution in exactly the same time — same ω — but the outer point has to cover far more distance to do it, so it must move faster.