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Lagrange of a rolling hoop without slipping

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A hoop of mass M and radius 1? rolls without slipping along a track which has the shape of a circle with radius 4R. It is subject to gravity. It is confined to a plane, so when the no-slip constraint is imposed there is just one degree of freedom.

Use the angle d as your coordinate. (This angle gives the location of the center of the hoop,
as measured from the center of the track circle.) Find T, U, L, and the Lagrange equation.
For small oscillations find the angular frequency of oscillation c..'. (Hint: Think carefully about
what the no-slip condition says about the rate of rotation of the hoop compared to the rate of
movement of its center, given by d. Remember you will have two terms in the kinetic energy -
which you can assume are due to the translational of its center of mass, and the rotation about its center of mass.

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I have shown very detailed step by step solution to this classical mechanics problem using Lagrange mechanics concepts.

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===============================================================================

------- Diagram -----------------

Constrain of motion is
R = 4R
R α = 4R θ + C

Where C is the ...

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