Answer Happy

8 [2] Consider a circular hoop of mass m and radius r, hanging from a massless spring with spring constant k and equilibrium length lo = 0 that is attached at the other end to the ceiling, let that attachment point be the origin. Assume that the spring remains straight but that its attachment point at the origin is free to pivot. This system has 6 degrees of freedom-the Cartesian coordinates corresponding to the center of mass of the hoop, R = (x, y, z), and the orientation of the hoop, described by the Euler angles. Let 2 lie in the vertical direction, opposite to the direction of gravity. Note that the angular momentum of the body frame relative to the space frame is w = - singé,' +ėéz' + (+ $cose)ez. R (c) (10 points) Find the angular momentum associated with motion of the center of mass with respect to the origin, in the lab frame.
(b) (10 points) Find the angular momentum associated with motion around the center of mass.
10 (c) (10 points) Find the kinetic energy in the lab frame in terms of time derivatives of the 6 degrees of freedom.
11 (d) (10 points) Choose & to point from the center of mass of the hoop to the point of contact on the hoop with the spring in the direction of 7.) and choose é to be normal to the plane of the hoop. Find the potential energy in the lab frame in terms of the coordinates (x, y, z) and the Euler angles.