Question

Define centripetal acceleration. Derive an expression for the centripetal acceleration of a particle moving with uniform speed $'V'$ along a circular path of radius $'r'$ . Give the direction of this acceleration.

Answer 1

To find the centripetal acceleration of the particle, substitute the centripetal force formula in Newton's second law of motion. The simplification of the above substituted equation, provides the relation for the centripetal acceleration.

Useful formula:
(1) The formula of the centripetal force is given by
$F = \dfrac{{m{v^2}}}{r}$
Where $F$ is the centripetal force, $m$ is the mass of the body, $v$ is the velocity of the body that undergoes centripetal motion and $r$ is the radius of the circular path.
(2) The Newton’s second law of motion is given by
$F = ma$
Where $a$ is the acceleration of the body.

Complete step by step solution:
The centripetal acceleration is defined as the rate of change of the tangential velocity. The centripetal force is the force that makes the body rotate in the circular path whose force acts inwards towards the centre of the rotation.
Using the formula of the centripetal force,

$F = \dfrac{{m{v^2}}}{r}$ -------------(1)

Let us consider the Newton’s second law of motion,

$F = ma$

By rearranging the parameters in the above equation, we get

$a = \dfrac{F}{m}$

Substituting the above equation, in the newton’s law of motion, we get

$a = \dfrac{{\dfrac{{m{v^2}}}{r}}}{m}$

By simplifying the terms,

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

Hence the centripetal acceleration of the body is obtained as $\dfrac{{{v^2}}}{r}$ . And this direction of the acceleration is towards inwards since it is centripetal force.

Note: Let us see the examples of the centripetal force, the rotation of the moon in the earth as the centre by keeping the earth’s gravitational force. The girl rotates the rope tied with the stone at its end, turning off the car along the road etc.