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 get 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. In circular motion, at that time the particle will have both centripetal and tangential acceleration as shown in the figure .
The resultant acceleration is given by vector sum of centripetal and tangential acceleration.
Since centripetal acceleration is ` a = (v^(2))/(r)` . The magnitude of this resultant acceleration is
given by `a_(R) = sqrt(a_(1)^(2)+((v^(2))/(r))^(2))`
This resultant acceleration makes an angle `theta` with the radius vector. It is given by
`tan theta = (a_(1))/(((v^(2))/(r)))`