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HW4 - Physics 110 Problem set#4 Due 1 A pendulum is...

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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 𝜙𝜙 is 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 (mass m ) is constrained to move along a stiff parabolic wire described by α y z = on the yz plane, as shown. a. Write down the Lagrangian of the system, using z y x , , as the generalized coordinates (2 pts). b. What are the 2 constraint equations? Express them in the form of 0 ; 0 2 1 = = G G (2 pts). c. Find the equations of motion for z y x , , , and the Lagrange multiplier(s) (4 pts). d. The same wire is spinning around the z axis 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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