Question

Relation between centripital acceleration and angular velocity

Answer 1

Centripetal Acceleration

Consider an object moving in a circle of radius r with constant angular velocity. The tangential speed is constant, but the direction of the tangential velocity vector changes as the object rotates.

Definition: Centripetal Acceleration

Centripetal acceleration is the rate of change of tangential velocity:

$\displaystyle\vec{a}_{c}^{}$ = $\displaystyle\lim_{\Delta t\to 0}^{}$$\displaystyle{\Delta \vec{v}_t \over \Delta t}$ (17)

Note:

The direction of the centripital acceleration is always inwards along the radius vector of the circular motion.

The magnitude of the centripetal acceleration is related to the tangential speed and angular velocity as follows:

ac = $\displaystyle{v_t^2 \over r}$ = $\displaystyle\omega^{2}_{}$r. (18)

In general, a particle moving in a circle experiences both angular acceleration and centripetal accelaration. Since the two are always perpendicular, by definition, the magnitude of the net acceleration a total is:

a total = $\displaystyle\sqrt{a_c^2 + a_t^2}$ = $\displaystyle\sqrt{r^2\omega^4 +

r^2\alpha^2}$. (19)

Definition: Centripetal Force

Centripetal force is the net force causing the centripetal acceleration of an object in circular motion. By Newton's Second Law:

$\displaystyle\vec{F}_{c}^{}$ = m$\displaystyle\vec{a}_{c}^{}$. (20)

Its direction is always inward along the radius vector, and its magnitude is given by:

Fc = mac = m$\displaystyle{v_t^2 \over r}$ = m$\displaystyle\omega^{2}_{}$r. (21)