For uniform circular motion, centripetal force is given by $$\dfrac{mv^2}{r}.$$ But what will be the centripetal force if the circular motion is non-uniform in the sense that linear velocity is changing its magnitude? Will the above relation still be valid for this case?
Answer
If the tangential velocity is changing in magnitude, that implies a tangential acceleration, and thus a tangential force in addition to the centripetal force.
If the motion of the object is in a circle of constant radius, then the instantaneous centripetal force is given by the expression you wrote.
The argument is not restricted to motion in a circle, but the analysis is easier for circular motion. For any point on any curved path one can find the circle that is tangent to the curve at that point. The instantaneous "centripetal" (perpendicular to the velocity) force is given by the same formula, with $r$ being the radius of the tangent circle (the "osculatory" or "kissing" circle).
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