(a) Apply energy conservation.
Kbottom = Up + Kp
½ mvbot2 = mghp + Kp
½3m (vo/3)2 = 3mg(r) + Kp
Kp = mvo2/6 – 3mgr
(b) The minimum speed to stay in contact is the limit point at the top where Fn just becomes zero. So set Fn = 0 at the top of the loop so that only mg is acting down on the block. Then apply Fnet(C)
Fnet(C) = mv2/r 3mg = 3m v2/r v = √rg
(c) Energy conservation, top of loop to bottom of loop
Utop + Ktop = Kbot
mgh + ½m vtop2 = 1/2mvbot2
g(2r) + ½(√rg)2 = ½(vo’)2
vo' = √5gr