What's the Difference Between Linear Speed and Angular Velocity in Circular Motion?

Linear speed is the distance travelled per unit time along a circular path, whereas angular velocity is the angle swept out per unit time. They are linked by $v=\omega r$, so linear speed also depends on the radius of the circular path.

This is part of A.2 Forces and momentum and applies to both SL and HL.

For one complete revolution, the object travels a distance $2\pi r$ while sweeping through an angle of $2\pi$ radians. Dividing both quantities by the time period $T$ gives

$$v=\frac{2\pi r}{T}, \qquad \omega=\frac{2\pi}{T}.$$

Therefore,

$$v=\omega r.$$

For example, a point $0.50\,\text{m}$ from the centre of a rotating wheel with $\omega=4.0\,\text{rad s}^{-1}$ has

$$v=(4.0)(0.50)=2.0\,\text{m s}^{-1}.$$

A point at $1.0\,\text{m}$ has the same angular velocity but twice the linear speed: $4.0\,\text{m s}^{-1}$.

Angular velocity also links to frequency and period: $\omega=2\pi f=2\pi/T$. Because radians are dimensionless, $\text{rad s}^{-1}$ is dimensionally equivalent to $\text{s}^{-1}$, but retaining “rad” clearly identifies angular motion.

Do not confuse either quantity with centripetal acceleration. Even when $v$ and $\omega$ are constant in uniform circular motion, the velocity vector changes direction, producing inward acceleration $a=v^2/r=\omega^2r$.

A common misconception is that every point on a rotating object has the same linear speed. In fact, all points have the same $\omega$, but points farther from the axis travel a larger circumference in the same time and therefore have a greater $v$.

Exam technique: When asked to calculate or determine a circular-motion quantity, identify whether the information is linear or angular, convert revolutions to radians if necessary, and use $v=\omega r$. Do not confuse linear speed with velocity: velocity changes direction continuously, even when speed is constant.