# Uniform circular motion

Uniform circular motion is the motion of an object moving at constant speed along the circumference of a circle. Although speed does not change, direction is constantly changing, so there must be acceleration.

As you can see, the velocity of the object is always a tangent to the circle. The object never flies away because it is always *accelerating towards the centre* of the circle. This acceleration is caused by a **centripetal force**.

We can relate the magnitude of centripetal acceleration to the object’s speed, the circle’s radius, and the period (time to complete one revolution) or frequency (revolutions per second) with a few equations:

${a}_{\text{c}}=\frac{{v}^{2}}{r}=\frac{4{\pi}^{2}r}{{T}^{2}}=4{\pi}^{2}r{f}^{2}\text{.}$

## Example

A machine can withstand a maximum centripetal acceleration of 235 m/s^{2}. If the radius is 13.6 cm, what is the maximum speed?

We can rearrange the first equation to get

$v=\sqrt{{a}_{\text{c}}r}=\sqrt{(235\text{}{\text{m/s}}^{2})(0.136\text{}\text{m})}=5.6533\text{}\text{m/s}\text{,}$

therefore the maximum speed is 5.65 m/s.