# 6.1.3 Centripetal Acceleration

Circular motion, as the name suggests, is motion along a circular path.

If the circular motion has a constant speed, then it is called uniform circular motion.

In circular motion, we often talk about the “tangential” and “radial” directions. Tangential refers to the direction parallel to the velocity, while radial refers to the direction perpendicular to the velocity.

It is most crucial to realize that the acceleration of a body in uniform circular motion in not zero. Even though the speed is constant, the velocity is continuously changing since its direction is continuously changing. So $\displaystyle a=\frac{{dv}}{{dt}}$ cannot be zero!

But we know that there is no acceleration along the tangential direction because the speed is constant. This means that that the acceleration must only be in the radial direction!

Because the radial direction points towards the center of the circle, this acceleration is commonly called the centripetal acceleration.

The magnitude of the centripetal acceleration is related to the speed and radius of the circular motion (read Appendix 6.A for the derivation) by the formula

$\displaystyle {{a}_{c}}=\frac{{{{v}^{2}}}}{r}$

Since $v=r\omega$, we can also write it as

${{a}_{c}}=r{{\omega }^{2}}$

Animation

Velocity and Acceleration during Circular Motion

Concept Test

1011