Figure shows a an incline which ends into a circular track of radius R. What should be the minimum value of height (h), so that the small object shown after release, is able to complete the loop. Neglect friction.

Explanation

It completes the loop, if its speed is atleast `sqrt(gR)` at B and `sqrt(5gR)` at A. Applying work energy theorem, for motion from starting point to the point B,
`mg(h-2R)=(1)/(2)m[(sqrt(gR))^(2)-0]`
`implies h=(5R)/(2)`

Alternatively : Applying work - energy theorem for motion from starting point to A.
`mgh=(1)/(2)m[(sqrt(5gR))^(2)-0]`
`impliesh=(5R)/(2)`