Centripetal Acceleration (HL IB Physics)

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Calculating Centripetal Acceleration

Centripetal acceleration is defined as:

The acceleration of an object towards the centre of a circle when an object is in motion (rotating) around a circle at a constant speed

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

Centripetal acceleration is always directed toward the centre of the circle, and is perpendicular to the object’s velocity

It is directed towards the centre of the circle as it is in the same direction as the centripetal force
It can be defined using the radius r and linear speed v:

$a = \frac{v^{2}}{r}$

Where:
- a = centripetal acceleration (m s^–2)
- v = linear speed (m s^–1)
- r = radius of the circular orbit (m)
Using the equation relating angular speed ω and linear speed v:

$v = r ω$

Where:
- ω = angular speed (rad s^–¹)

These equations can be combined to give another form of the centripetal acceleration equation:

$a = ω^{2} r$

Alternatively, since we know angular velocity is:

$ω = 2 πf = \frac{2 π}{T}$

Where:
- f = frequency (Hz)
- T = time period (s)
This means the centripetal acceleration can also be written as:

$a = {(\frac{2 π}{T})}^{2} r = \frac{4 π^{2} r}{T^{2}}$

This equation shows how the centripetal acceleration relates to the linear speed and the angular speed

Worked example

A ball tied to a string is rotating in a horizontal circle with a radius of 1.5 m and an angular speed of 3.5 rad s⁻¹.

Calculate its centripetal acceleration if the radius was twice as large and angular speed was twice as fast.

WE - Centripetal acceleration answer image, downloadable AS & A Level Physics revision notes

Exam Tip

The equations for centripetal acceleration are given on your data sheet in multiple forms. Which form you use depends on what you're given in the question i.e. v or ω

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Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.