IIT (NTSE/Olympiad)  

Circular Motion & Rotational Dynamics

7. Centripetal Acceleration and centripetal Force

7. Centripetal Acceleration and centripetal Force
(i) A body or particle moving in a curved path always moves effectively in a circle at any instant.
(ii) The velocity of the particle changes moving on the curved path, this change in velocity is brought by a force known as centripetal force and the acceleration so produced in the body is known as centripetal acceleration.
(iii) The direction of centripetal force or acceleration is always towards the centre of circular path.
7.1 Expression for Centripetal Acceleration

This is the magnitude of centripetal acceleration of particle
(i) It is a vector quantity. In vector form = ×
(ii) The direction of would be the same as that of Δ
(iii) Because velocity vector at any point is tangential to the circular path at that point, the acceleration vector acts along radius of the circle at that point and is directed towards the centre. This is the reason that it is called centripetal acceleration.
7.2 Expression for Centripetal force
If v = velocity of particle, r = radius of path

Then necessary centripetal force
Fc = mass × acceleration
Fc = mv2/r
This is the expression for centripetal force

(i) It is a vector quantity
(ii) In vector form

(iii) For circular motion : || = m (vω sin 90°) = mvω
Note :
1. Centripetal force is not a real force. It is only the requirement for circular motion.
2. It is not a new kind of force. Any of the forces found in nature such as gravitational force, electric friction force, tension in string reaction force may act as centripetal force.

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