Circular Motion

Uniform circular motion. When a point object moves on a circular path at constant speed, covering equal arc lengths in equal time intervals, it is said to be in uniform circular motion. Even though the speed (the magnitude of $\vec v$) never changes, the direction of $\vec v$ is continuously turning — velocity is always tangent to the circle, so as the particle goes around, its velocity vector sweeps through every direction in the plane of the circle.

Centripetal acceleration. Because the velocity vector's direction is changing even though its magnitude is not, there must be an acceleration, even in uniform circular motion — it is called centripetal acceleration, and it always points toward the centre of the circle ('centripetal' literally means 'centre-seeking').

Derivation: let the particle be at position $\vec r_1$ with velocity $\vec v_1$, and a short time $\Delta t$ later at position $\vec r_2$ with velocity $\vec v_2$; for uniform circular motion $|\vec r_1|=|\vec r_2|=r$ and $|\vec v_1|=|\vec v_2|=v$. In this small interval, both the position vector and the velocity vector turn through the same small angle $\theta$ (since $\vec v\perp\vec r$ at every instant, whatever angle $\vec r$ turns through, $\vec v$ turns through the identical angle). This makes the triangle of $\vec r_1,\vec r_2,\Delta\vec r$ geometrically similar to the triangle of $\vec v_1,\vec v_2,\Delta\vec v$, giving the proportion

and, because $\Delta\vec v$ points radially inward (toward the centre) in this construction,

Dividing by $\Delta t$ and taking the limit $\Delta t\to0$,

which has magnitude

and, using $v=r\omega$ from §2.11.5, this can equally be written $a_c=\omega^2r$. Either form is standard. The centripetal acceleration has constant magnitude throughout uniform circular motion, but its direction is constantly changing (always toward the instantaneous centre), so as a vector it is very much not constant.

Non-uniform circular motion. If the object's speed also changes as it goes around (e.g. a pendulum bob swinging in a vertical circle is faster at the bottom than near the top), the motion is non-uniform circular motion, and the particle then has both a centripetal acceleration $a_c=v^2/r$ (from the changing direction) and a tangential acceleration $a_t$ (from the changing speed, as derived in §2.11.5) acting simultaneously. The resultant acceleration is the vector sum of these two mutually perpendicular pieces:

making an angle $\theta$ with the radius vector (i.e. with $a_c$) given by

$\tan\theta = \frac{a_t}{a_c} = \frac{a_t}{v^2/r}$ …