Chapter 8, Problem 8.7P

A point mass is constrained to move on a massless hoop of radius a fixedin a vertical plane that rotates about its vertical symmetry axis with constantangular speed ω. Obtain the Lagrange equations of motion assuming the only external forces arise from gravity. What are the constants of motion? Show that if ω is greater than a critical value ω0, there can be a solution in which the particle remains stationary on the hoop at a point other than at thebottom, but that if ω < ω0, the only stationary point for the particle is atthe bottom of the hoop. What is the value of ω0?

Verify that each of the following force fields is conservative. Then find, for each, a scalar potential φ such that F = −∇φ.(1). F = i − zj − yk.

. Consider a system of N particles in a uniform gravitational field. Prove that the total gravitational torque about center of mass(CM) is zero.?​