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:

$$\vec{a}_{c}^{} \lim_{\Delta t\to 0}^{} {\Delta \vec{v}_t \over \Delta t}$$ (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:

  $${v_t^2 \over r} \omega^{2}_{}$$ (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:

  $$\sqrt{a_c^2 + a_t^2} \sqrt{r^2\omega^4 + r^2\alpha^2}$$ (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:

$$\vec{F}_{c}^{} \vec{a}_{c}^{}$$ (20)

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

$${v_t^2 \over r} \omega^{2}_{}$$ (21)