Calculating Centripetal Acceleration

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

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

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:

$v = r ω$

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

$a = \frac{{(r ω)}^{2}}{r}$

$a = r ω^{2}$

This equation shows that centripetal acceleration is equal to the radius times the square of the angular speed
Alternatively, rearrange for r:

$r = \frac{v}{ω}$

This equation can be combined with the first one to give us another form of the centripetal acceleration equation:

$a = \frac{v^{2}}{(\frac{v}{ω})}$

$a = v ω$

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

Where:
- a = centripetal acceleration (m s⁻²)
- v = linear speed (m s⁻¹)
- ⍵ = angular speed (rad s⁻¹)
- r = radius of the orbit (m)

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.

Answer:

Step 1: State acceleration in terms of angular speed

$a = r ω^{2}$

Step 2: Increase angular acceleration with twice the radius and twice the angular speed

$a = (2 r) \times {(2 ω)}^{2} = 2 r \times 4 ω^{2} = 8 r ω^{2}$

Step 3: Substitute in the known values to calculate

$a = 8 r ω^{2} = 8 \times 1.5 \times 3 . 5^{2} = 147 m s^{- 2}$

Calculating Centripetal Acceleration (CIE A Level Physics)