Motion in a circle

12.1 Kinematics of uniform circular motion

The radian

An angle in radians is arc length divided by radius. One radian is the angle at the centre of a circle subtended by an arc equal in length to the radius.

A full circle has arc length $2\pi r$, so a complete turn is $2\pi$ rad. So $360° = 2\pi$ rad, and $1\text{ rad} \approx 57°$.

Angular speed

Angular speed $\omega$ is the angle swept out by the radius per unit time, in rad/s.

$\omega = \frac{2\pi}{T} = 2\pi f$

In time $t$, an object moves through an arc $s = r\theta$, so its linear speed is:

$v = r\omega$

12.2 Centripetal acceleration

Why there is an acceleration

An object in a circle at constant speed has a constantly changing direction, so its velocity changes. It therefore accelerates, and there must be a resultant force.

• A force of constant size, always perpendicular to the velocity, changes the direction but not the speed.
• It points towards the centre of the circle, so we call it the centripetal force.
• It does no work, because it is at right angles to the motion, so the kinetic energy stays constant.

The equations

$a = r\omega^2 = \frac{v^2}{r}$

$F = mr\omega^2 = \frac{mv^2}{r}$

Solving problems

Identify which real forces act, find their resultant towards the centre, and set that equal to $mv^2/r$.

At the bottom of a vertical circle, tension acts up towards the centre and the weight acts away, so $T - mg = mv^2/r$. This makes the tension greatest at the bottom. At the top, the string goes slack if $v^2 < rg$.

Banked turns and conical pendulums

For a conical pendulum, or an aircraft banking, the force at an angle $\theta$ has two jobs:

• The vertical component balances the weight, $T\cos\theta = mg$.
• The horizontal component provides the centripetal force, $T\sin\theta = mv^2/r$.

Dividing gives $\tan\theta = v^2/(rg)$, which is independent of the mass.