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Centripetal acceleration

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Summary

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

DESCRIPTION

EQUATION

Centripetal acceleration
​a=v2ra=ω2ra=ωva = \frac{v^2}{r} \newline \\[0.1in]a=\omega^2 r \newline \\[0.1in]a = \omega va=rv2​a=ω2ra=ωv​​​​

 

VARIABLE DEFINITIONS

​​quantity name

symbol

derived unit

si BASE units

​radiusradiusradius​​
​rrr​​
​mmm​​
​mmm​​
​linear velocitylinear\ velocitylinear velocity​​
​vvv​​
​ms−1ms^{-1}ms−1​​
​ms−1ms^{-1}ms−1​​
​angular velocityangular\ velocityangular velocity​​
​ω\omegaω​​
​rads−1rads^{-1}rads−1​​
​rads−1rads^{-1}rads−1​​
​centripetal accelerationcentripetal \space accelerationcentripetal acceleration​​
​aaa​​
​ms−2ms^{-2}ms−2​​
​ms−2ms^{-2}ms−2​​


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.
1
Velocity
2
Centripetal acceleration


Centripetal acceleration is given as: 


​a=v2ra = \frac{v^2}{r}a=rv2​    (1)


Substituting v=ωrv=\omega rv=ωr into equation (1):


  a=(ω r)2ra = \dfrac{(\omega \ r)^2}{r} \newline a=r(ω r)2​


a=ω2r2ra=ω2ra = \dfrac{\omega^2 r^2}{r} \newline \newline \\[0.1in] a=\omega^2 ra=rω2r2​a=ω2r​​


Substituting v=ωr→r=vωv=\omega r \rightarrow r=\frac v \omegav=ωr→r=ωv​ into equation (1):


a=v2(vw)a=ωva = \dfrac{v^2}{(\dfrac{v}{w})} \newline \\[0.1in]a=\omega va=(wv​)v2​a=ωv​​


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−13.7\ rads^{-1}3.7 rads−1. The distance between the yo-yo and finger is 38 cm38\ cm38 cm. Calculate the centripetal acceleration of the yo-yo. 


Firstly, write down the known values: 


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


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


a=ω2ra = {\omega}^2 ra=ω2r


Then, substitute the values into the equation:


a=3.72×0.38a=5.2022a = 3.7 ^2 \times0.38 \newline \\[0.1in]a=5.2022a=3.72×0.38a=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−2centripetal \ acceleration, \ a=5.2\ ms^{-2}centripetal acceleration, a=5.2 ms−2​​​​​


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

​

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FAQs - Frequently Asked Questions

Where is the centripetal acceleration of an object directed?

The centripetal acceleration of an object is directed towards the centre of a circle.

What is centripetal acceleration?

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

Why does an object in circular motion accelerate?

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.

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