Spring 2007 1 Mech

Consider an Atwood machine with a massless pulley and two masses, m and M, which are attached at opposite ends to a string of fixed length that is hung over the pulley. For this Atwood machine, the center of the pulley is supported by a spring of spring constant k.

(a) Find the Lagrangian and the resulting equations of motion.

(b) Find the equilibrium position of the pulley and its frequency of oscillations. Consider your result in the limits m = M and discuss.
Hint: using the method of Langrange multipliers significantly reduces the complexity of this problem.




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