rigid body, cylinder

A spool of thread has rolled under the bed! Fortunately, you get the free end of the thread. The inner part of the reel where the thread is wound is a homogeneous, full cylinder of mass 3M and radius 2R. At each end there is a homogeneous cylindrical disc with mass M and radius 3R. The total mass of reels m = 5M. The moment of inertia of a homogeneous, full cylinder about its axis of symmetry is: I = 1 / 2mr ^ 2. The free end of the thread comes out from the underside of the inner cylinder, in the middle between the two outer discs. There is friction between the reel and the floor, with a static coefficient of friction = 0.25 and a dynamic coefficient of friction = 0.2.

1. Show that the maximum force at which the reel rolls without slipping is:
F <F_max = 5 / 3Mg

2. In which direction does the reel move as you pull with the force F = F_max? Do you get out of bed? Explain.

The next thing you do is to give the thread a strong bounce in the horizontal direction, that is, you use a large horizontal force F over a very short time interval t. The force is much greater than the frictional force, and you can therefore disregard the frictional force in that time interval while the string is working. After this time interval, when the force F has stopped working, the reel moves at speed v = v_o and angular velocity w_0.

3. How big is the impulse that the spool gets?

4. Show that the angular velocity w_0 is related to the linear velocity v_0 as follows:

w_o = w_0 ^ k = 2/3 (v_0 / R) ^ k