Problem $4(80$ points$)$: A rectangular slab of mass $m_S=4kg$ moves across a series of rollers, each of which is initially at rest and equivalent to a uniform disk of mass $m_R=2$ kg $.$ Since the length of the slab is slightly less than three times the distance $b$ between two adjacent rollers, the slab leaves a roller just before it reaches another one. Each time a new roller enters into contact with the slab, slipping occurs between the roller and the slab for a short period of time $($less than the time needed for the slab to move through the distance $b).$ Knowing that the velocity of the slab in the position shown is $v_0=3\frac{m}{s}$ detarmine helnars

a$)(10$pts$)$ Draw the impulse$-$momentum diagram for the roller B or C$.$

b$)(10$pts$)$ Determine the moment of the center of the roller B or C$.$

c$)(10$pts$)$ Draw the impulse$-$momentum diagram for the slab.

d$)(10$pts$)$ Determine the velocity of the slab after it has moved a distance $b.$

e$)(10$pts$)$ Determine the velocity of the slab after it has moved a distance $nb.$

f$)(10$pts$)$ Determine the initial kinetic energy of the whole system.

g$)(10$pts$)$ Determine the terminal kinetic energy of the whole system when the slab completely stops.

h$)(10$ pts