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### 2.5 Motions in circle(basics cntd..)

Here we'll discuss the centripetal force, centripetal acceleration and the motion caused by the central force. Before we proceed ahead please review the previous section here.

We'll first discuss qualitatively motion in a curved path due to a perpendicular force.

Above diagram show the qualitative motion. We know that according to the Newton's first law of motion when a particle is freely moving, it moves in a straight line at uniform speed unless acted upon by an external force. If incase force is applied perpendicular to the direction of travel, it would produce an acceleration in that direction.

There are three important things to remember,
• speed of the particle does not change because the acceleration is perpendicular to the velocity.
• the direction of the velocity, however will be changed by the acceleration and it results in the curved path.
• the direction of the curve is in the direction of the acceleration and the acceleration points towards the centre of the curvature. The path will be curved so as long the perpendicular force is applied.
Now consider the following diagram below,

Centripetal acceleration: The magnitude of the centripetal acceleration is given by the following expression.
$$a=r\omega ^{2}=\frac{v^{2}}{r}$$
where, $\omega$ = angular velocity, in rad $s^{-1}$
$v$ =linear velocity, in $ms^{-1}$and
$r$ = radius of the circular motion, in $m$

Centripetal force: If there is an acceleration then it must be that according to Newton second law, there's a force. In this case it is a centripetal force and it is defined as " a force which acts on a body moving in a circular path and is directed towards the centre around which the body is moving"

The magnitude of the centripetal force is given by,
$$F=ma=m\frac{v^{2}}{r}$$
where, $m$ = mass of an object, in $kg$
and $a$= centripetal acceleration, in $ms^{-2}$

Exercise: Watch this video of problems on circular motion.