Consider a hoop of mass 3.0 kg and radius 0.50 m that rolls without slipping down an incline. The hoop starts at rest from a height h = 2.5 m above the bottom of the incline and then rolls to the bottom. 

(a) What forces act on the hoop? Which of these forces do work on the hoop? 

(b) Is the mechanical energy of the hoop conserved? 

(c) Make a sketch showing the hoop at the top and the bottom of the incline. 

(d) What is the total initial mechanical energy of the hoop? Take the zero of potential energy to be at the bottom of the incline. 

(e) If the hoop’s speed at the bottom of the incline is v_f, what is its mechanical energy? Express your answer in terms of v_f.