Doubt from work, power and energy

#1

please solve ques 5

#2

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#3

The question is incorrect.
The value of the distance comes out to be \frac{am}{\sqrt{M^2-m^2}} and not \frac{2amM}{{M^2-m^2}}. Here's the solution. Please do check.

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#4

You are incorrect. Acceleration = 0 does not mean that body comes to rest, it simply means that there is no increment in the velocity, which happens to be + ve. So the ring stays in motion.

Sure, at the distance you derived, a=0, but v is not equal to 0. This can be shown by applying the work energy theorem.

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