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

Explanation

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 vertical circle, the speed of the bob is not the same at all time. Whenever the speed is not same in circular motion, the particle will have both centripetal and tangential accelerastion a shown in the figure.

The resultant acceleration is obtained by vector sum of centripetal and tangential acceleration.
Since centripetal acceleration `(v^(2))/(r)`. is the magnitude of
this resultant acceleration is given by `a_(R)=sqrt(a_(t)^(2)+((v^(2))/(r))^(2))`
This resultant acceleration makes an angle `theta` with the radius vector as shown in figure. This angle is given by `tantheta=(a_(t))/((v^(2)//r))`