Result

Centripetal acceleration and force

a_{c} = \frac{v ^{2}}{r}, F_{c} = \frac{m v ^{2}}{r}

Statement

An object moving in a circle of radius r at speed v accelerates toward the centre with this magnitude; the net force producing it is the centripetal force.

Symbols and units

$a_{c}$	centripetal acceleration	m/s²
$v$	speed	m/s
$r$	radius of the path	m
$F_{c}$	centripetal force	N
$m$	mass of the object	kg

When it applies

Circular motion at constant or varying speed; a_c is only the component of acceleration toward the centre.

Don't use when

Not a distinct physical force — always identify which real force (tension, gravity, friction, normal force) is providing it.

Derivation

For uniform circular motion the velocity vector rotates at constant rate while its magnitude stays fixed. Differentiating position twice for circular motion gives an acceleration directed inward with magnitude:

a_{c} = \frac{v ^{2}}{r} = ω^{2} r

Newton's second law then gives the net force required to sustain it:

F_{c} = m a_{c} = \frac{m v ^{2}}{r}

Worked examples

Minimum speed at the top of a loop

At the top of a vertical loop, gravity alone must supply the centripetal force at minimum speed:

m g = \frac{m v ^{2}}{r} \Rightarrow v_{m i n} = g r

For r = 5.0 m:

v_{m i n} = (9.8) (5.0) \approx 7.0 m/s

Watch out for

Centripetal force is not a new, separate kind of force — it is the name given to whatever net force (tension, gravity, friction, normal force) happens to point toward the centre.

Connects to

Newton's second law