Question

Derive the expression for total acceleration in the non-uniform circular motion.

Answer 1

If the speed of the object in circular motion is not constant, then we have non-uniform circular motion. For example, when the bob attached to a string moves in a vertical circle, the speed of the bob is not the same at all times. Whenever the speed is not the same in a circular motion, the particle will have both centripetal and tangential acceleration as shown in the figure.

Resultant acceleration (a_R) in a non-uniform circular motion 

The resultant acceleration is obtained by the vector sum of centripetal and tangential acceleration Since centripetal acceleration is ${\displaystyle \frac{v^2}{r}}$, the magnitude of this resultant acceleration is given by ${\displaystyle -a_R=\sqrt{a_t^2+{\left(\frac{v^2}{r}\right)}^2}}$

This resultant acceleration makes an angle 0 with the radius vector as shown in the figure.
This angle is given by tan θ = ${\displaystyle \frac{a_t}{\left(v^2\text{/}r\right)}}$