Physics

A body is allowed to slide down a frictionless track freely under gravity.The track ends in a semicircular shaped part of diameter d.What should be the height(minimum) from which the body must fall so that it completes the circle?

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  1. Assuming the "semicircular shape" is like the loop of a roller-coaster, and we want the body to complete the circle.

    At the top of the loop, of diameter d (radius d/2), the velocity must be such that the centripetal force exerted by the rails ≥0, or
    mv²/(d/2)-mg ≥0

    At this position, the kinetic energy is (1/2)mv^2=mgh, where h is the height of release of the body above the top of the loop.
    Substitute in above, we find:
    2mgh/(d/2)-mg ≥0
    or simply
    h > d/4.
    So the body must be released at a point h> d/4+d, above the lowest part of the loop, or
    h > 5d/4

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  2. Anything falling from a height of h will rise to a height of h so h = d Period

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