Note: Put your calculator into radians mode before using circular motion equations!
Remember Newton's First law?
"If an object continues in a straight line at constant velocity, all forces acting on the object are balanced."
Or another way of putting it...
"An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force."
Objects moving in circular motion clearly aren't going in a straight line so the forces can't be balanced.
There is a resultant force. This is called the centripetal force.
The centripetal force is always directed towards the centre of the circle (along the radius of the circle).
If an object is moving with constant speed in circular motion, it is not going at constant velocity. That's because velocity is a vector. Although its magnitude remains the same, its direction varies continuously.
This resultant force, the centripetal force, causes the centripetal acceleration. It is always at 90^{ο} to the direction of movement of the object - and that's why the object doesn't speed up!
Centripetal acceleration can be calculated using:
Where:
v = velocity (m/s)
a = centripetal acceleration (m/s^{2})
r = radius of the circle (m)
And from Newton's Second Law:
F = ma, so
This is an equation for centripetal force.
When an object is moving in circles that are vertical, its weight has to be taken into consideration.
Note: If you are using v = ω r in your syllabus, you can substitute this into the equations for centripetal force and acceleration to find values using angular velocity.
The radius of a circle and its circumference are related by the equation...
Circumference = 2πr
As long as you use angles in radians you can write this general equation:
s = rθ
Where:
s = arc length covered
r = radius of circle
θ = angle in radians
In linear or straight-line motion, we measure speed by looking at how much distance is covered each second. You can do that in circular motion too, but it's often better to use angular speed, ω.
Angular speed measures the angle of a complete circle (measured in radians) you cover per second.
For instance,
Where:
θ = angle in radians
t = time taken in seconds.
If you consider that the time taken for a complete rotation is the period, T, then
because 2 is the angle covered (in radians) when you do a complete circle.
Remembering that you can also write this as
ω = 2πf
If you are going round in a circle of radius, r, and you are travelling at a linear speed, v ms^{-1}:
The distance covered in 1 rotation = 2πr
The time for one rotation = T, the period.
Linear speed
So, v = 2&rf and w = 2pf
or
= w and v = wr
So we can relate angular and linear speed.
ω = 2πf
v = rω
ω = angular speed, rad s^{-1}
f = frequency, Hz (No. of rotations per second)
T = the period of rotation, s
v = linear speed, ms^{-1}
r = radius of rotation, m
a = centripetal acceleration, ms^{-2}
F = centripetal force, N