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.

Topics
Energy conservation, Mechanical energy, Friction, Kinematics, Newtonian mechanics, Classical mechanics, Gravitational force, Functions and functionals, Differential geometry, Books
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When the surface is concave upwards, b̂·ẑ=1 and the remain-on-the-surface condition, N(s)≥0⁠, implies that v2(s)/gR≥H(s)⁠. Therefore, the initial speed must satisfy v02/gR≥H(0)⁠. Because g·n̂(0)<0 in this case, we have H(0)≤0 and therefore v02/gR≥H(0) 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.

11.

When φ is a decreasing function (the curve is concave upwards) we have κ=−dφ/ds and (b̂·ẑ)=1⁠, so that v2(φ)=v02e2μ(φ−φ0)−2ge2μφ∫φ0φ(sin φ′−μ cos φ′)e−2μφ′κ(φ′) dφ′⁠.

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