Centripetal Force — F = mv²/r

The centripetal force formula F = mv²/r explained: why circular motion needs an inward force, and a worked car-cornering example.

F = \frac{mv^2}{r}

The inward force needed for circular motion equals mass times speed squared divided by radius.

What it means

Anything moving in a circle is accelerating toward the center — even at constant speed — because its velocity direction keeps changing. Centripetal force is the name for whatever net inward force causes that acceleration: friction for a cornering car, tension for a whirling ball on a string, gravity for an orbiting satellite. The $v^2$ dependence is the striking part: doubling the speed demands four times the inward force, which is why fast corners are treacherous. There is no outward 'centrifugal force' in an inertial frame — what you feel pressed against the car door is your body's inertia resisting the inward acceleration.

Variables

Symbol	Name	Unit	Meaning
$F$	Centripetal force	N	Net inward force causing circular motion
$m$	Mass	kg	Mass of the circulating object
$v$	Speed	m/s	Tangential speed along the circle
$r$	Radius	m	Radius of the circular path

Assumptions

Speed is constant (uniform circular motion)
Circular path of fixed radius r
Net force considered — individual forces like friction provide it

Derivation

In uniform circular motion the velocity vector rotates at angular speed $\omega = v/r$ . The rate of change of velocity is $a = v\omega = v^2/r$ directed toward the center. Newton's second law then gives $F = ma = mv^2/r$ .

Worked example

Problem

A 1000 kg car takes a flat curve of radius 50 m at 15 m/s. What centripetal force keeps it on the curve?

Given:

m = 1000 kg
v = 15 m/s
r = 50 m

Write the formula: $F = mv^2/r$
Substitute: $F = 1000 \times 15^2 / 50$
Calculate: $15^2 = 225$ , so $F = 225{,}000 / 50 = 4500\ \mathrm{N}$

4500 N toward the center of the curve

The tires must supply 4500 N of inward grip — about 0.46 g of lateral acceleration. If the road is wet and friction cannot provide it, the car slides outward: not because a force pushes it out, but because nothing pulls it in.