A small car is let go from rest on a smooth track at a height h, and runs round a vertical loop of…

On a curved path, use energy for the speed and $F = \dfrac{mv^2}{r}$ toward the centre for the forces. At the top of a loop the track's push $N$ and the weight both point down: $N + mg = \dfrac{mv^2}{R}$, and the cart stays on while $N \ge 0$.

When only conservative forces (gravity, springs) do work, mechanical energy is conserved: $K_1 + U_1 = K_2 + U_2$. Forces that do no work, like a smooth surface's push or a string at right angles to the motion, do not spoil it.