A mass \(M_{1}\) is hung on an unstretchable string A, and the other end of string A is passed over a fixed, frictionless, non-rotating pulley \(P_{1}\), as shown. This other end of string A is then attached to the center of a second frictionless, non-rotating pulley \(P_{2}\) of mass \(M_{2}\), over which is passed a second nonstretchable string \(\mathrm{B}\), one end of which is attached to a hanging mass \(m_{1}\) while the other end is attached to a hanging mass \(m_{2}\). Let \(X_{1}(t)\) be the length of string A beneath the center of pulley \(P_{1} ; X_{2}(t)\) be the length of string A beneath the center of \(P_{1} ; x_{1}(t)\) be the length of string \(\mathrm{B}\) beneath the center of pulley \(P_{2}\); and \(x_{2}(t)\) be the length of string \(\mathrm{B}\) beneath the center of pulley \(P_{2}\). There is a uniform gravity \(g\) downward.

(a) Find the total kinetic energy of the system, in terms of \(X_{1}, x_{1}\), and the various masses.

(b) Find the total potential energy of the system, measured from the center of fixed pulley \(P_{1}\).

(c) Find the Lagrangian of the system.

(d) Find the acceleration of mass \(M_{1}\) in terms of given quantities.