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?

Answer 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

Answer 2

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