Loop the Loop

At the top of a loop the track and the weight both push the cart down, toward the centre: $\relax N + mg = mv^2/r$.

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

$$v_{\min} = \sqrt{g r}$$

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

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