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A small block of mass 3m moving at speed v0/3 enters the bottom of the circular, vertical loop-the-loop shown, which has a radius r. The surface contact between the block and the loop is friction less. Determine each of the following in terms of m, vo, r, and g.

a. The kinetic energy of the block and bullet when they reach point P on the loop

b. The speed vmin of the block at the top of the loop to remain in contact with track at all times

c The new required entry speed vo' at the bottom of the loop such that the conditions in part b apply. 

1 Answer

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Best answer

(a) Apply energy conservation. 

Kbottom = Up + K

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 F= 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

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