Consider the modified Atwood machine shown in the figure below. The two weights on the left...

  

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Consider the modified Atwood machine shown in the figure below. The two weights on the left have equal masses m and are connected by a massless spring of Hooke's-law constant k. The weight on the right has mass M = the pulley is massless and frictionless. The coordinate x is the extension of the spring from its equilibrium length; that is, the length of the spring is le + x, where le is the equilibrium length (with all the weights in position and M held stationary). (a) Show that the total potential energy (spring plus gravitational) is just U two momenta conjugate to x and y. Solve for i and ý, and write down the Hamiltonian. Show that the coordinate y is ignorable. (c) Write down the four Hamilton equations an solve them for the following initial conditions: You hold the mass M fixed with the whole system in equilibrium and y = Yo. Still holding M fixed, you pull the lower mass m down a distance o, and at t = 0 you let go of both masses. [Hint: Write down the initial values of x, y, and their momenta. You can solve the r equations by combining them into a second-order equation for r. Once you know r(t), you can quickly write down the other three variables.] Describe the motion. In particular, find the frequency with which r oscillates. 2m, and kx? (plus a constant that we can take to be zero). (b) Find the y M=2m m le +x Consider the modified Atwood machine shown in the figure below. The two weights on the left have equal masses m and are connected by a massless spring of Hooke's-law constant k. The weight on the right has mass M = the pulley is massless and frictionless. The coordinate x is the extension of the spring from its equilibrium length; that is, the length of the spring is le + x, where le is the equilibrium length (with all the weights in position and M held stationary). (a) Show that the total potential energy (spring plus gravitational) is just U two momenta conjugate to x and y. Solve for i and ý, and write down the Hamiltonian. Show that the coordinate y is ignorable. (c) Write down the four Hamilton equations an solve them for the following initial conditions: You hold the mass M fixed with the whole system in equilibrium and y = Yo. Still holding M fixed, you pull the lower mass m down a distance o, and at t = 0 you let go of both masses. [Hint: Write down the initial values of x, y, and their momenta. You can solve the r equations by combining them into a second-order equation for r. Once you know r(t), you can quickly write down the other three variables.] Describe the motion. In particular, find the frequency with which r oscillates. 2m, and kx? (plus a constant that we can take to be zero). (b) Find the y M=2m m le +x