Physics 110, Problem set #4 Due April 28, 2010 1.A pendulum is suspended from the edge of a rotating disk, as shown. a.Using (𝜃𝜃,𝜙𝜙,𝜓𝜓)as the generalized coordinates, write down the Lagrangian of the system (10 pts). Note that 𝜙𝜙is measured from the projection of 𝜓𝜓, as shown. b.What are the conserved quantities? There are at least two (6 pts). c.Find three equations of motion for the system (6 pts). *This problem is not hard at all with Mathematica.* `2.Constraints (25 pts): A bead with (massm) is constrained to move along a stiff parabolic wire described by αyz=on the yzplane, as shown. a.Write down the Lagrangian of the system, using zyx,,as the generalized coordinates (2 pts). b.What are the 2 constraint equations? Express them in the form of 0;021==GG(2 pts). c.Find the equations of motion forzyx,,, and the Lagrange multiplier(s) (4 pts). d.The same wire is spinning around thezaxis with angular speed ω(counterclockwise). Repeat parts (b)(c) (6 pts). e.Use these equations to find the equilibrium position of the bead. (4 pts)
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