From what height, h, should a rail-cart fall to complete a loop-the-loop of radius r without falling off a the track? Assume the track on which the rail-cart travels is smooth and express h in terms of r.

This question combines the principles of energy conservation and circular motion. The insight is to realise the condition for which the rail-cart will fall off of the track - when there is no contact force between the cart and track at the apex of the loop.Energy of cart at top of track: mgh Energy of cart at apex of loop: mg(2r) + 0.5mv²Hence energy conservation implies that mgh = mg(2r)Newton's Second Law at apex of the loop: F_centripetal = mv²/r = mg - F_contactSetting F_contact = 0 and substituting v² = g into energy conservation equation gives h = 2.5r