Centripetal Force

What centripetal force is

A body moving in a circle is continuously changing direction. Since velocity is a vector, a change in direction is a change in velocity — even if speed is constant. Therefore a force must be acting.

This force is directed toward the centre of the circle and is called centripetal force:

$F_c = \frac{mv^2}{r} = mr\omega^2$

where $v$ is the speed, $r$ is the radius, $\omega$ is the angular velocity, and $m$ is the mass.

Derivation of centripetal acceleration

Consider a particle moving in a circle of radius $r$ at constant speed $v$.

At time $t$, velocity is $\vec{v}_1$. At time $t + \Delta t$, velocity is $\vec{v}_2$. Both have magnitude $v$ but different directions.

The particle has moved through a small angle $\Delta\theta$ in time $\Delta t$.

The change in velocity $\Delta\vec{v} = \vec{v}_2 - \vec{v}_1$.

For small $\Delta\theta$, the magnitude of $\Delta\vec{v}$:

$|\Delta\vec{v}| = v \cdot \Delta\theta$

(since the velocity vector of magnitude $v$ has rotated by $\Delta\theta$)

The direction of $\Delta\vec{v}$ points toward the centre (for small $\Delta\theta$, the chord direction approaches the radius direction).

Acceleration:

$a = \lim_{\Delta t \to 0} \frac{|\Delta\vec{v}|}{\Delta t} = \lim_{\Delta t \to 0} \frac{v \cdot \Delta\theta}{\Delta t} = v \cdot \frac{d\theta}{dt} = v\omega$

Since $v = r\omega$:

$a = v\omega = \frac{v^2}{r} = r\omega^2$

This acceleration is directed toward the centre — it is called centripetal acceleration.

By Newton's Second Law:

$\boxed{F_c = ma_c = \frac{mv^2}{r} = mr\omega^2}$

"Centripetal force" is not a new force

This is crucial: centripetal force is not a separate type of force. It is the name given to whatever force (or combination of forces) provides the centripetal acceleration.

In each case, identify the actual physical force and set it equal to $\frac{mv^2}{r}$.

What centrifugal force is

In an inertial frame, there is no centrifugal force. The only real force is centripetal — toward the centre.

In a rotating (non-inertial) frame, an observer feels a pseudo force directed outward — this is called centrifugal force. It is not a real force; it is a consequence of being in a non-inertial frame.

The passenger in a car turning a corner feels pushed outward — this is the centrifugal effect. In reality, the car seat is pushing them inward (centripetal), and their inertia makes them feel the outward push relative to the car.

Effect of speed and radius

$F_c = \frac{mv^2}{r}$

• Double the speed → quadruple the centripetal force needed
• Double the radius → halve the centripetal force needed

This is why sharp turns (small $r$) at high speed are dangerous — the required centripetal force grows very rapidly.