Circular Motion

Circle Geometry

In Higher Physics, all motion was seen to be linear (acting in a straight line). All equations of motion used up to this point have also been linear. To describe circular motion, the equivalent rotational equations of motion must be found. 

To understand the motion of objects in a circular path, the basic geometry of a circle is required:-

The diagram above shows the main parts of a circle, with the following parts:-

Circumference - The length of the perimeter of a circle.

Arc - The length of a section of the circumference of a circle.

Diameter - The length of the line connecting opposite sides of a circle through its centre.

Radius - The length of the line connecting the edge of a circle to its centre.

Tangent - A line at a right angle to the radius that 'touches' the edge of a circle. 

Measurements within a Circle 

There are two main methods for calculating angles within a circle:-

Degrees - 360° in one full circle.

Radians - 2π radians in one full circle.

         1 radian = 57.3°

The video below shows an introduction to "radians", and how to convert between radians and degrees:-

Note - Throughout the Advanced Higher course, any rotational working must be calculated in radians. This means that your calculator must be in "Rad" mode.

Angular Displacement

The first variable to be considered is the rotational equivalent of displacement, the angular displacement (θ), which is defined as the change in angular position between two points and can be found by:-

Where :-

θ  =  The angular displacement (in radians) 

s  =   The arc length 

r  =  The radius of the circle 

To convert between linear and angular displacement, the following formula can be used:-

 s  =  r θ

Angular Velocity 

An object moving in a circular path will have an Angular Velocity ( ω ), which is defined as the rate of change of Angular Displacement and can be found by:-

Where:-

ω  =  Angular velocity (radians s-1)

θ  =  Angular displacement (radians) 

t   =  time (s) 

To convert between linear (tangential ) velocity and angular velocity for an object, the following formula can be used:-

v  = r ω

Alternate versions of Angular Velocity

The above formula and working give the S.I. units of angular velocity (rad s-1); however, there are two other useful methods of describing angular velocity:-

Periodic time - Time taken for 1 complete revolution (s) 

Revolutions per second (1 / Periodic Time) 

The periodic time is related to the angular velocity by:-

Angular Acceleration

An object which changes its angular velocity will have an angular acceleration (α), which is defined as the rate of change of angular velocity and can be found by:-

Where:-

α  =  Angular acceleration (radians s-2) 

ω  =  Angular velocity (radians s-1) 

θ  =  Angular displacement (radians) 

t  =  time (s) 

In order to convert between linear and angular velocity for an object, the following formula can be used:-

a  =  r α

Example 1 -

A child's Roundabout of diameter 3 m rotates at a rate of 0.75 revolutions every second. What is the tangential velocity of a child sitting at the edge of the Roundabout? 

1 Revolution = 2π radians 

0.75 Rev s-1 = 3/2π radians s-1 

v  =  r ω

v  =  1.5 x 3/2π

v  =  7.1 ms-1 

Note - When completing calculations, use the fractional values of π throughout; only convert to a decimal for the final answer. 

Equations of Angular Motion 

As can be seen from the above formulae, there is a linear relationship between the tangential and angular velocities. Due to this, the equations of motion can be easily converted into angular motion:-

Where:-

θ    =  Angular displacement (radians) 

ω0 =  Initial angular velocity (radians s-1) 

ω  =  Final angular velocity (radians s-1) 

α    =  Angular acceleration (radians s-2) 

t     =  time (s) 

Example 2 - 

A particle initially at rest begins to move in a circle. After a time of 20 seconds, the particle has an angular Velocity of 16π rad s-1. 

Calculate:-

1. The angular acceleration of the particle.

2. The number of complete revolutions the particle makes.

θ    =  ?

ω0 =  0  

ω  =  16π radians s-1 

α    =  ?

t     =  20 s 

Angular acceleration :-

ω  =  ω0 + αt 

16π  =  0 + ( α x 20 )

α  =  4/5π

α  =  2.51 radians s-2 

Angular displacement :-

θ  = ω0t  + 1/2αt2

θ  =  0  +  ( 1/2 x ( 4/5π x 202 ) )

θ  =  160π radians 

Convert between angular displacement and revolutions:- 

1 Revolution = 2π radians

Number of revolutions = 160π / 2π 

Number of revolutions =  80