# Sliding or Rolling of a Sphere, Cylinder, and Tube down a Semicircular Well

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This Demonstration shows the motion of a solid sphere, a solid cylinder, and a thin-walled cylindrical tube inside a semicircular well. All three bodies have the same radius and are released simultaneously from the rim of the well with zero initial velocity.

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Contributed by: Erik Mahieu (November 2011)

Open content licensed under CC BY-NC-SA

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## Details

The moments of inertia are for a thin-walled tube, for a solid cylinder, and for a solid sphere.

The Lagrangian for a rolling body, with radius and mass , in a semicircular well with radius is given by .

If we substitute (where is 1, 1/2, or 2/5) into the equation of motion, it becomes and the mass cancels out of the equation.

See also the Demonstrations "Disk Sliding or Rolling in a Semicircular Well" by Sarah Lichtblau and "Moment of Inertia" by Bernard Vuilleumier.

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