## Summary

We have all experienced circular motion. Here are some common thoughts among students...

If your speed is constant how can you be accelerating?

Velocity is a vector quantity. Anything travelling in a circular path is changing direction and, therefore, accelerating. A resultant force is required.

When you move in a circle it feels like you are being thrown out. So how do you stay in your seat?

The forces in circular motion act towards the centre of the circle. The sensation of being 'flung outwards' comes from your inertia relative to the system.

If too many people join me on a roller coaster will we all fall to our death on the loop the loop?

This would affect the total mass of the body, which in turn affects the kinetic energy required to make it round. Take a look at the following pages to find out!

## Get the basics

#### Basic quantities

Rather than considering displacement, velocity and time, in this topic we will consider the angular equivalents. It is helpful to recall that circular motion is *periodic*, which means it repeats in cycles.

**Time period (T)** = time for one complete circle. Unit: s

**Frequency (f) **= number of complete circles per second (\(1\over T\)). Unit: Hz (s^{-1})

**Angular displacement (θ)** = angle swept out in radians. Unit: rad

**Angular velocity (ω)** = angle swept out per unit time. Unit: rad s^{-1}

This could be a good time to revisit Radians.

Small angle approximation: Angle in radians = arc length/radius (\(s \over r\)). However, if the angle is small this is approximately the same as the chord length/radius.

#### Speed and angular velocity

**Speed** (distance / time) is related to the angular velocity.

\(v={2\pi r \over T}=\omega r\)

Note that the units are now satisfactory (it's ok to ignore radians).

#### Centripetal force

The **centripetal force** is the *resultant force *acting towards the centre that makes a body travel in a circle.

\(F=m \omega^2 r={mv^2 \over r}\)

Note that, if you are asked to name the force that is acting on an object, it will not be the 'centripetal force'. This is simply the consequence of the combination of the other forces at play.

In everyday speech, we might refer to a 'centrifugal force' as flinging objects outward. In fact, this does not exist and instead is an effect of remaining temporarily stationary while the system turns the corner. Relative motion does not always require a force!

## Test yourself

*Check out this πg physics summary.*

*Use flashcards to practise your recall.*