A body starts from rest at the top of an inclined plane of height h and enters a vertical circular track of radius R.

Find the minimum value of h required so that the body just completes the vertical circle.

For a body to just complete a vertical circle, the normal reaction at the highest point must be zero.

At the topmost point:

mV²/R = mg

Therefore:

V² = gR

The minimum kinetic energy at the top becomes:

K.E.

= ½mV²

= ½mgR

The topmost point of the circle is at a height:

2R

above the bottom of the track.

Applying conservation of mechanical energy:

Initial Potential Energy

= Final Potential Energy + Final Kinetic Energy

mgh

= mg(2R) + ½mgR

= (5/2)mgR

Cancelling mg:

h = 5R/2

This is the minimum height required for successful completion of the loop without losing contact.

Final Answer

h = 5R/2