Calculating Centripetal Force

An object moving in a circle is not in equilibrium, it has a resultant force acting upon it
- This is known as the centripetal force and is what keeps the object moving in a circle
The centripetal force (F) is defined as:

The resultant force towards the centre of the circle required to keep a body in uniform circular motion. It is always directed towards the centre of the body's rotation.

Centripetal force can be calculated using:

$F = \frac{m v^{2}}{r} = m r ω^{2} = m v ω$

Centripetal Force

Centripetal force diagram, downloadable AS & A Level Physics revision notes

Centripetal force is always perpendicular to the direction of travel

Where:
- F = centripetal force (N)
- v = linear velocity (m s^-1)
- ⍵ = angular speed (rad s^-1)
- r = radius of the orbit (m)

Note: centripetal force and centripetal acceleration act in the same direction
- This is due to Newton’s Second Law
The centripetal force is not a separate force of its own
- It can be any type of force, depending on the situation, which keeps an object moving in a circular path

Examples of Centripetal Force

Situation	Centripetal Force
Car travelling around a roundabout	Moving objects have energy in their kinetic store
Ball attached to a rope moving in a circle	Objects gain energy in their gravitational potential store when they are lifted above ground
Earth orbiting the Sun	Objects have energy in their elastic potential store if they are stretched

Worked example

A bucket of mass 8.0 kg is filled with water is attached to a string of length 0.5 m.

What is the minimum speed the bucket must have at the top of the circle so no water spills out?

WE - Centripetal force question image, downloadable AS & A Level Physics revision notes

Answer:

Step 1: Draw the forces on the bucket at the top

Step 2: Calculate the centripetal force

The weight of the bucket = mg
This is equal to the centripetal force since it is directed towards the centre of the circle

$m g = \frac{m v^{2}}{r}$

Step 3: Rearrange for velocity v

m cancels from both sides

$v = \sqrt{g r}$

Step 4: Substitute in values

$v = \sqrt{9.81 \times 0.5} = 2.21 m s^{- 1}$

Calculating Centripetal Force (CIE A Level Physics)

Revision Note