A stone tied to a string and whirled overhead, the tip of a fan blade, a satellite circling the Earth — all move along a circular path. When an object moves in a circle at constant speed, we call it uniform circular motion. The word uniform refers to the speed, not the velocity: even though the speed never changes, the direction of motion changes at every instant, so the velocity changes — which means the object is always accelerating.
Instead of measuring distance along the arc, it is easier to measure the angle swept at the centre. This is the angular displacement $\theta$, measured in radians (one full circle is $2\pi$ radians). The rate at which this angle is swept is the angular velocity $\omega=\dfrac{\theta}{t}$, with SI unit radian per second ($\text{rad/s}$). The ordinary (linear) speed $v$ along the circle of radius $r$ is linked to angular velocity by the simple and important relation $v=\omega r$.
Circular motion repeats itself, so we describe it with two more quantities. The time period $T$ is the time for one complete revolution, and the frequency $f$ is the number of revolutions per second, with $f=\dfrac{1}{T}$ (unit hertz, $\text{Hz}$). In one period the object sweeps $2\pi$ radians, so $\omega=\dfrac{2\pi}{T}=2\pi f$. These let us convert easily between revolutions, angles and time.
The most important idea is the acceleration. Because the velocity continuously changes direction, there is an acceleration directed at every instant towards the centre of the circle. This is the centripetal acceleration $a_c=\dfrac{v^2}{r}=\omega^2 r$. The word centripetal means centre-seeking. The force that produces it — tension in the string, gravity for a satellite, friction for a car turning — is the centripetal force $F_c=\dfrac{mv^2}{r}=m\omega^2 r$, always directed towards the centre. Without it, the object would fly off along a tangent in a straight line, exactly as Newton's first law predicts.
- Uniform circular motion: constant speed, continuously changing direction (so the velocity changes).
- Angular velocity: $\omega=\dfrac{\theta}{t}$, unit rad/s; linked to speed by $v=\omega r$.
- Period and frequency: $f=\dfrac{1}{T}$ and $\omega=\dfrac{2\pi}{T}=2\pi f$.
- Centripetal acceleration: $a_c=\dfrac{v^2}{r}=\omega^2 r$, directed towards the centre.
- Centripetal force: $F_c=\dfrac{mv^2}{r}=m\omega^2 r$, also towards the centre.