A 8 kg block of ice, released from rest at the top of a 1.5 m long frictionless ramp, slides downhill, reaching a speed of 2.5 m/s at the bottom. What is the angle between the ramp and the horizontal? What would be the speed of the ice at the bottom if the motion were opposed by a constant friction force of 10 N parallel to the surface of the ramp?

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.