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What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R so that it can complete the loop?

1. √(gR)
2. √(2gR)
3. √(3gR)
4. √(5gR)

Correct answer: √(5gR)

Solution

To complete the vertical loop, the body must have sufficient velocity at the top of the loop to maintain contact with the track. At the top, the centripetal force is provided by the gravitational force and the normal force. For minimum velocity, the normal force becomes zero, and the centripetal force equals the gravitational force. Using energy conservation and centripetal force conditions, the minimum velocity at the bottom is derived as √(5gR).

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