Circular Motion

Physics: AQA A Level, topic 6.1.1

Radians

A radian is another unit for measuring angles. There are $2\pi$ radians in a full circle. The radian unit can be written as rad.

Definition of a radian

The angle between two radii of a circle connected by an arc length that is equal in length to the radius is equal to one radian.

Comparison to degrees

There are 2 $\pi$ radians and 360 degrees in a circle.
If an object travels all the way around a circle it has done one revolution.
$1\text{ rad}=\frac{360}{2\pi} \approx 57.3^o$

Change in angle

For a circle of radius $r$ , the change in angle is equal to the distance moved along the circumference divided by the radius.
The distance along the circumference moved is $\Delta s$ .
This can be written as $\Delta\theta = \frac{\Delta s}{r}$

Angular Speed

Angular speed measures how quickly an object is rotating. The units of angular speed are radians per second (rad s^-1).

Calculating angular velocity

Angular velocity is the change in angle of an object over a period of time. The equation for calculating angular velocity is:
Angular velocity = change in angle ÷ change in time
$\omega = {\Delta\theta} / {\Delta}t$

Comparison to linear velocity

Linear velocity is when an object is travelling in a straight line:
$v={\Delta}s/{\Delta}t$
The radian equation helps us compare linear and angular velocity:
${\Delta}s=r{\Delta\theta}$
By substituting one into the other, we get the following equation:
$v=r{\omega}$
From this equation, we know that linear velocity is proportional to the distance from the centre, r.

Period of Circular Motion

In circular motion, the period and frequency are linked to the angular speed. These are helpful quantities to know if we need to calculate the angular speed.

Period and frequency

In circular motion, the period of an object is how long it takes to travel all the way around the circle.
In circular motion, the frequency of an object is how many times it goes around the circle in one second.
Period = 1 ÷ frequency of the object
$T=1\;{\div}\;f$

Calculating angular speed

The equation for calculating angular speed is:
Angular velocity = 2 × pi × frequency of the object
${\omega}=2{\pi}\; {\times}\; f$
By using $T=1\;{\div}\;f$ , we get:
${\omega}=2{\pi}\;{\div}\;T$

Test yourself on Circular Motion

How many radians make a full turn?
What is π radians in degrees?
What is the correct equation for angular speed?
How does the linear velocity of an object relate to the angular velocity?
What is the period of an object moving around a circle?

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