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.[more]
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 .[less]
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.