A particle in uniform circular motion (UCM) moves in a circle or circular arc at constant linear speed v. The instantaneous linear velocity \(\vec{v}\) of the particle is along the tangent to the path in the sense of motion of the particle. Since \(\vec{v}\) changes in direction, without change in its magnitude, there must be an acceleration that must be

1. perpendicular to \(\vec{v}\)

2. constant in magnitude 

3. at every instant directed radially inward, 

i.e., towards the centre of the circular path.

Such a radially inward acceleration is called a centripetal acceleration.

∴ \(\vec{a}\) = \(\frac{d \vec{v}}{dt}\) = \(\vec{a_r}\)

If \(\vec{\omega}\) is the constant angular velocity of the particle and r is the radius of the circle,

\(\vec{a_r}\) = -\(\omega^2\vec r\)

where ω = | \(\vec{\omega}\) | and the minus sign shows that the direction of \(\vec{a_r}\) is at every instant opposite to that of the radius vector \(\vec{r}\). In magnitude,

\(a_r\) = \(\omega^2r\) = \(\frac{v^2}r\) = \(\omega v\)

[Note : The word centripetal comes from Latin for ‘centre-seeking’.]