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 in nature. In order to describe Circular Motion, the equivalent Rotational Equations of Motion must be found.
In order to understand the Motion of Objects in a Circle, first 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 center
Radius - The Length of the Line connecting the Edge of a Circle to its center
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
In order 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 )
In order 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 gives 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 is Angular Velocity will have an Angular Acceleration (), 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 been 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 in to Angular motion :-
