Two lumps of clay moving in opposite directions collide with each other and move off as one with no external forces exerted on the system. The first lump has a mass of 0.40 kg and a velocity of +2.0 m/s and the other a mass of 1.6 kg and a velocity of -1.0 m/s. What is the velocity of the center of mass before the collision?

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.