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Question
Derive the expression for total acceleration in the non-uniform circular motion.
Solution
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 (aR) in a non-uniform circular motion
The resultant acceleration is obtained by the vector sum of centripetal and tangential acceleration Since centripetal acceleration is `v^2/r`, 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 0 with the radius vector as shown in the figure.
This angle is given by tan θ = `a_t/((v^2"/"r))`
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