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/s². If the radius is 13.6 cm, what is the maximum speed?

We can rearrange the first equation to get

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

therefore the maximum speed is 5.65 m/s.