The motion of a block sliding on a curve is a well studied problem for flat and circular surfaces, but the necessary conditions for the block to leave the surface deserve a deeper treatment. In this article, we generalize this problem to an arbitrary surface, including the effects of friction, and provide a general expression to determine under what conditions a particle will leave the surface. An explicit integral form for the speed is given, which is analytically integrable for some cases. We demonstrate general criteria to determine the critical speed at which the particle immediately leaves the surface. Three curves, a circle, a cycloid, and a catenary, are analyzed in detail, revealing several interesting features.
When the surface is concave upwards, and the remain-on-the-surface condition, , implies that . Therefore, the initial speed must satisfy . Because in this case, we have and therefore for all concave-upwards surfaces. The end result is that the initial speed has no restrictions—the particle always stays on the surface if the surface is concave upwards.
When is a decreasing function (the curve is concave upwards) we have and , so that .