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1. A particle of mass m is constrained to slide without friction along a straight rod. The rod is rotating in a vertical plane about one end with a constant angular velocity w, in the presence of a uniform gravitational field g. The rod is horizontal at t = 0, and the particle is at rest a distance d from the fixed end of the rod. g m Fixed End w A particle rotating in a vertical plane along a straight rod. a (a) Find the Lagrangian and the differential equations of motion for the particle. (b) Show that there is a certain critical angular frequency we such that, if w <wc, the particle will eventually hit the fixed end of the rod and, if w > wc, the particle will eventually fly off the rotating end of the rod. Find wc.