Slipping on an Arbitrary Surface with Friction

The motion of a block slipping on a surface is a well studied problem for flat and circular surfaces, but the necessary conditions for the block to leave (or not) the surface deserve a detailed treatment. In this article, using basic differential geometry, we generalize this problem to an arbitrary surface, including the effects of friction, providing a general expression to determine under which conditions the particle leaves the surface. An explicit integral form for the velocity is given, which is analytically integrable for some cases, and we find general criteria to determine the critical velocity at which the particle immediately leaves the surface. Five curves, a circle, ellipse, parabola, catenary and cycloid, are analyzed in detail. Several interesting features appear, for instance, in the absense of friction, a particle moving on a catenary leaves the surface just before touching the floor, and in the case of the parabola, the particle never leaves the surface, regardless of the friction. A phase diagram that separates the conditions that lead to a particle stopping in the surface to those that lead to a particle departuring from the surface is also discussed.