When an object moves in a circle, its speed may stay the same but its direction changes.
The change in direction is described by an acceleration that always points toward the centre of the circle.
This is called centripetal acceleration (the word “centripetal” means “toward the centre”).
The magnitude of centripetal acceleration is given by:
\$a_c = \frac{v^2}{r} = \omega^2 r = \frac{v \, \omega}{1}\$
where:
Imagine a ball tied to a string and swung around in a circle.
The tension in the string pulls the ball toward the centre.
Even though the ball’s speed stays the same, its direction changes continuously.
Because force is the cause of acceleration (Newton’s 2nd law), the constant‑magnitude force that is always perpendicular to the velocity produces a centripetal acceleration that keeps the ball moving in a circle.
On a carousel, each seat is attached to a central pole by a rail.
As the carousel spins, the rail pushes the seat toward the centre.
The push is always perpendicular to the seat’s motion, yet it keeps the seat moving in a circle.
That push is the centripetal force, and the resulting change in direction is the centripetal acceleration.
A car is driving around a circular track with a radius of 50 m at a constant speed of 20 m s⁻¹.
What is the centripetal acceleration?
Force is the cause; acceleration is the effect.
Newton’s 2nd law links them: \$Fc = m ac\$.
If a car of mass 1500 kg has \$a_c = 8\,\text{m s}^{-2}\$, the required centripetal force is:
\$F_c = 1500 \times 8 = 12\,000\,\text{N}.\$
| Quantity | Symbol | Formula |
|---|---|---|
| Centripetal acceleration | \$a_c\$ | \$\displaystyle a_c = \frac{v^2}{r}\$ |
| Centripetal force | \$F_c\$ | \$\displaystyle Fc = m ac = \frac{m v^2}{r}\$ |
| Angular speed | \$\omega\$ | \$\displaystyle \omega = \frac{v}{r}\$ |