Uniform Circular Motion

Uniform circular motion is motion in which a particle travels along a circular path of

fixed radius $r$ at a constant speed $v$. Although the speed never changes, the velocity

is continuously changing, because velocity is a vector and its direction — always tangent to

the circle — keeps turning as the particle goes around. Since acceleration is defined as the

rate of change of the velocity vector (not just of speed), a particle in uniform circular

motion is always accelerating, even though it never speeds up or slows down.

Direction of the acceleration. Consider the particle at two nearby points on the circle,

separated by a short time interval, with velocity vectors $\vec v_1$ and $\vec v_2$ of equal

magnitude but slightly different direction (see the figure). The change $\Delta \vec v = \vec v_2 - \vec v_1$, found by the triangle law, points — in the limit of a very short time

interval — exactly toward the centre of the circle, perpendicular to the (instantaneous)

velocity. Since $\vec a = \Delta \vec v/\Delta t$, the acceleration of a particle in uniform

circular motion is always directed radially inward, toward the centre; this is why it is

called centripetal ("centre-seeking") acceleration.

Magnitude. The magnitude of the centripetal acceleration is

where $\omega$ is the angular velocity of the particle, $\omega = v/r$, measured in

radians per second and equal to the rate $d\theta/dt$ at which the angle swept out at the

centre increases with time. The period $T$, the time for one complete revolution, and

the frequency $f$, the number of revolutions per second, are related to $\omega$ by

It is worth stressing the word uniform: if the particle's speed along the circular path

were also changing with time, there would be an additional acceleration component directed

tangent to the circle (a tangential acceleration), on top of the centripetal component — …