Centripetal Acceleration (DP IB Physics: SL): Revision Note

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 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:

$v=r\omega$

• Where:

  • ω = angular speed (rad s^–1)

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

$a={\omega }^{2}r$

• Alternatively, since we know angular velocity is:

$\omega =2\mathrm{\pi }f=\frac{2\mathrm{\pi }}{T}$

• Where:

  • f = frequency (Hz)

  • T = time period (s)

• This means the centripetal acceleration can also be written as:

$a={\left(\frac{2\mathrm{\pi }}{T}\right)}^{2}r=\frac{4{\mathrm{\pi }}^{2}r}{T^2}$

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