A hoop of mass M and radius R 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 freedorn. Use the angle θ 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 frequcncy of oscillation ω.
(Hint: Think carefuily 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 θ. Remember you will have two terms in the kinetic energ'y - which you can assume are due to the trarrslation of its center of mass, and the rotation about its center of mass.)
