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homework7

# homework7 - 1 CDS 140a Homework Set 7 Due Wednesday...

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1 CDS 140a: Homework Set 7 Due: Wednesday, December 2, 2009. For the first three questions, consider the mechanical system shown in the Figure below. The mass M slides (without friction) along the curve y = x 2 . The mass m hangs by a light rod of length l (as a planar pendulum) from the mass M . Let θ denote the angle with the vertical, as shown. Both masses are subject to a downward gravitational force. M m y x g ; gravity θ y = x 2 1. Write down a Lagrangian for the system and compute the Euler–Lagrange equations. 2. Does this system have a conserved energy? Explain how to get this expression in both the Lagrangian and the Hamiltonian formulations of the problem. 3. Note that when M is stationary at x = 0 and m is hanging vertically (with θ = 0), one has an equilibrium. Is it stable? Asymptotically stable? What if one adds friction? 4. The support point of a simple planar pendulum of mass m and length l moves along the horizontal x -axis with position x ( t ) = a cos ωt .

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