Sliding down an arbitrary curve in the presence of friction

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. (C) 2017 American Association of Physics Teachers.