A small heavy block is attached to the lower end of a light...

A small heavy block is attached to the lower end of a light rod of length $l$ which can be rotated about its clamped upper end. What minimum horizontal velocity should the block be given so that it moves in a complete vertical circle?

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Key Concepts

Centripetal Force in Circular Motion

For an object moving in a circular path, a net inward force, called the centripetal force, is required to keep the object moving along the curved path. This force is provided by tension, gravity, or a combination thereof, and its magnitude must be sufficient at all points in the motion to maintain the circular trajectory, particularly at the top of the circle where gravitational force plays a significant role.

Conservation of Energy

This concept involves the idea that the total mechanical energy (kinetic plus potential) in a system remains constant if only conservative forces are doing work. In problems involving circular motion, energy conservation allows us to relate the speed at different positions along the motion path by equating the loss in kinetic energy to the gain in potential energy as the object climbs, or vice versa.

Critical Speed at the Top of the Circle

At the highest point of a circular path, the tension in a rod or string can vanish if the object is moving at the minimum required speed. This critical speed is determined by setting the centripetal force equal to the gravitational force, yielding v = ?(g*l) for a circle of radius l. Achieving this speed ensures that the object can complete the circle without the tension in the connecting rod or string going slack.

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