Loop the Loop

At the top of a loop the track and the weight both push the cart down, toward the centre: N + mg = mv²/r.

At the minimum speed N = 0, and the mass cancels:

v_min = √(g r)

Any slower and the cart leaves the track. Starting from rest without friction, the start must be at least 2.5r high.

At the minimum speed the weight alone pulls the cart round the top of the loop.