Centripetal acceleration

In a nutshell

​Centripetal acceleration is perpendicular to the instantaneous velocity of an object moving in a uniform circular motion and is always directed towards the centre of a circle.  

EQUATIONS

VARIABLE DEFINITIONS

Centripetal Acceleration 

Acceleration is defined as the rate of change of velocity with respect to time. Velocity is a vector quantity meaning it has both a magnitude and a direction. 

If an object is moving in a circle, its direction is constantly changing, meaning the velocity is constantly changing. As acceleration is the rate of change of velocity, if the velocity is changing, then the object must also be accelerating. 

This is called the centripetal acceleration and is always directed towards the centre of the circle that the object is travelling in, which is perpendicular to the instantaneous velocity of the object.

Centripetal acceleration is given as: 

​$$a = \frac{v^2}{r}$$    (1)

Substituting $$v=\omega r$$ into equation (1):

  $$a = \dfrac{(\omega \ r)^2}{r} \newline$$

$$a = \dfrac{\omega^2 r^2}{r} \newline \newline \\[0.1in] a=\omega^2 r$$​​

Substituting $$v=\omega r \rightarrow r=\frac v \omega$$ into equation (1):

$$a = \dfrac{v^2}{(\dfrac{v}{w})} \newline \\[0.1in]a=\omega v$$​​

Note: Check which equations you are given in the exam booklet, you might have them all or you may only have one! 

​​Example

A yo-yo is swung around at an angular velocity of $$3.7\ rads^{-1}$$. The distance between the yo-yo and finger is $$38\ cm$$. Calculate the centripetal acceleration of the yo-yo. 

Firstly, write down the known values: 

$$\omega = 3.7\ rads^{-1} \newline \\[0.1in]r=0.38 \, m$$

​Next, write down the equations needed and rearrange if necessary:

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

Then, substitute the values into the equation:

$$a = 3.7 ^2 \times0.38 \newline \\[0.1in]a=5.2022$$

Make sure to include units and round to the lowest number of significant figures of the values given by the question:

$$centripetal \ acceleration, \ a=5.2\ ms^{-2}$$​​​​​

The centripetal acceleration of the yo-yo is $$\underline{5.2\ ms^{-2}}.$$​

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