A thin hoop of radius R and mass M oscillates in its own plane with one point of the hoop fixed. Attached to the hoop is a point mass M constrained to move without friction along the hoop. The system is in a uniform gravitational field g. Consider only small oscillations. (a) Show that the normal-mode frequencies are 1/2g 1/2 2 R W2 = 2g) 1/2 R (b) Find the normal-mode eigenvectors. Sketch the motion. (c) Construct the modal matrix. (d) Find the normal coordinates and show that they diagonalize the lagrangian.