Radian measure

An angle at the centre of a circle separates two radii. The part of the circle’s circumference between those radii is the arc subtended by the angle.

The angle in radians is the arc length divided by the radius:

\theta = \frac{s}{r}

Here, $\theta$ is the angle in radians, $$$s$$$ is the arc length and $$$r$$$ is the radius. The arc length and radius must be measured in the same units.

One radian is the angle subtended by an arc whose length equals the radius: when $$$s = r$$$ , $\theta = 1$ radian.

Radians and a complete revolution

A complete revolution follows the whole circumference, whose length is $2\pi r$ . Dividing this distance by the radius gives an angle of $2\pi$ radians. Therefore, $2\pi$ radians is equivalent to $360^\circ$ , and $\pi$ radians is equivalent to $180^\circ$ .

Radians and distance along a circular path

Rearranging the definition gives the distance travelled along an arc:

s = r\theta

This relationship uses $\theta$ in radians. For a given radius, a larger angular displacement corresponds directly to a longer distance along the circular path.

Angular speed

Angular speed describes how quickly an object turns around a point. It is the rate at which its angular displacement changes. Angular displacement is the angle through which the object has turned, measured in radians.

For motion at a constant angular speed, if the object turns through an angle $\Delta\theta$ in a time $\Delta t$ :

\omega = \frac{\Delta\theta}{\Delta t}

Angular speed $\omega$ is measured in radians per second ( $\mathrm{rad,s^{-1}}$ ). If the angular speed changes, this calculation gives the average angular speed over the time interval.

Angular speed and linear speed

For an object moving around a circle of radius $$$r$$$ , its linear speed $$$v$$$ is the distance it travels along the circular path per second. A larger angle corresponds to a longer distance along the path. For an angle measured in radians, that distance is the radius multiplied by the angle, so:

\omega = \frac{v}{r}

At the same linear speed, an object moving around a smaller circle has a greater angular speed: it turns through a larger angle each second.

Angular speed and frequency

The frequency $$$f$$$ is the number of complete revolutions per second. One complete revolution corresponds to an angle of $2\pi$ radians. Therefore, the angle turned through each second is $2\pi$ multiplied by the number of revolutions per second:

\omega = 2\pi f

For example, at one revolution per second, $f = 1,\mathrm{Hz}$ and the angular speed is $2\pi,\mathrm{rad,s^{-1}}$ .

Centripetal acceleration

An object moving around a circle at constant speed is accelerating, even though it is not speeding up or slowing down. Velocity includes direction as well as speed. As the object moves, its direction changes continuously, so its velocity changes continuously.

The acceleration caused by this change in direction is called centripetal acceleration. At every point on the circular path, it points towards the centre of the circle. The object's velocity points along the tangent to the circle, perpendicular to the centripetal acceleration.

Magnitude of centripetal acceleration

The magnitude of the acceleration can be calculated using either the object's speed or its angular speed:

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

Here, $$$a$$$ is the centripetal acceleration, $$$v$$$ is the speed, $$$r$$$ is the radius of the circular path and $\omega$ is the angular speed, which measures how quickly the object turns through an angle. The two forms are equivalent because $v = \omega r$ .

For a fixed radius, doubling the speed makes the centripetal acceleration four times as large. At constant speed, the acceleration has a constant magnitude, but its direction changes as the object moves around the circle.

Centripetal force

An object moving in a circle continually changes direction. Even if its speed stays constant, its velocity changes, so it accelerates. This centripetal acceleration is directed towards the centre of the circle.

A resultant force must act in the same direction as the acceleration. The centripetal force is this inward resultant force: it causes the change in direction needed for circular motion.

F = ma = \frac{mv^2}{r} = m\omega^2r

Here, $$$F$$$ is the centripetal force, $$$m$$$ is the object's mass, $$$a$$$ is its centripetal acceleration, $$$v$$$ is its speed, $$$r$$$ is the radius of the circle, and $\omega$ is its angular speed in radians per second. The two forms of the force expression are equivalent because $v = \omega r$ .

For the same mass and radius, a higher speed requires a greater inward force. For the same mass and speed, a smaller radius also requires a greater inward force.

Centripetal force is not an extra force. It names the role of the resultant of the forces acting towards and away from the centre. For example, tension in a string or gravity can provide the inward force, depending on the situation. These forces are already included when finding the resultant; a separate centripetal force must not be added.