# Disk Sliding or Rolling in a Semicircular Well

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The disk of radius in this Demonstration is acted upon by gravity, , and either slides or rolls in the semicircular well of radius . In reality it would only slide if the surface of the well were completely frictionless.

Contributed by: Sarah Lichtblau (January 2009)

Open content licensed under CC BY-NC-SA

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## Details

Developed from the teaching materials of Thomas Peacock and David Gossard for Course 2.003 in the Department of Mechanical Engineering at MIT.

For the most general case, the Lagrangian for the problem is given by where is the moment of inertia of the disk and Θ, its angle of rotation.

If the disk's motion is purely sliding, , and the equation of motion reduces to .

If the disk rolls without slipping, then and the equation of motion works out to .

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