1. All the ramps below are identical, smooth and 5 m high except for the shape of their inclines. The crates at the top of each ramp are identical, with a mass of 2 kg. Initially, stationary crates are released from the top of the ramps. What is the initial GPE of the crates in each case? Show and explain your work.

Consider a double atwood machine constructed as follows: a mass 4m is suspended from a string that passes over a massless pulley on frictionless bearings. the other end of this string supports a second similar pulley, over which passes a second string supporting a mass of 3m at one end and m at the other. using two suitable generalized coordinates, set up the lagrangian and use the lagrange equations to find the acceleration of the mass 4m when the system is released. explain why the top pulley rotates even though it carries equal weights on each side.