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Centripetal force and acceleration

The centripetal force is the force which is perpendicular to the direction of motion of an object undergoing circular motion.

It points to the centre of the circle, and is responsible for changing the direction of the velocity.

F is the centripetal force.
F is the centripetal force.

It is often confused with the centrifugal force, which is the reaction force that points outwards from the centre of rotation (opposite to the centripetal force).

The centrifugal force is considered to be a "fictitious force" as it only acts in the frame of reference of the object.

The centripetal force does not refer to a specific type of force like tension, friction or gravitation. The type of centripetal force will be different depending on the situation.

Situation Type of centripetal force
Planet orbiting sun Gravity
Ball swung on string Tension

A simple expression for the magnitude of centripetal force follows from Newton's second law:$$$\begin{align*}F=ma=\frac{mv^{2}}{r}=mr\omega^{2}\end{align*}$$$The variable $$a$$ is the magnitude of centripetal acceleration (i.e. acceleration directed towards the centre). It is given by:$$$\begin{align*}a&=\frac{v^{2}}{r}=r\omega^{2}\end{align*}$$$

$$v=$$linear speed; $$\omega=$$angular speed; $$r=$$radius of circular motion.

An object moving in uniform circular motion with angular displacement $$\vecphy{\theta}$$, angular velocity $$\vecphy{\omega}$$, speed $$\vecphy{v}$$ and radius $$r$$
An object moving in uniform circular motion with angular displacement $$\vecphy{\theta}$$, angular velocity $$\vecphy{\omega}$$, speed $$\vecphy{v}$$ and radius $$r$$