9860

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

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.
The oak well surface causes sufficient friction for the bodies to roll without slipping. The polished marble well surface, being nearly frictionless, causes the bodies to slide without rolling.
A rolling body has lower acceleration because its net motion is a combination of translation of the whole body down the well together with rotation about its axis. Therefore, the body with the largest moment of inertia (the thin-walled cylinder), will have the slowest acceleration. The three sliding bodies have only translational motion and slide down the well with the same acceleration.
The resulting equation of motion is .

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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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