A roller coaster car is released from rest at the top of the firstrise and then

Question

A roller coaster car is released from rest at the top of the firstrise and then moves freely with negligible friction. The rollercoaster as a circular loop of radius R in a vertical plane. a)Suppose first the car barely makes it around the loop; at the topof the loop, the riders are upside down and feel weightless. Findthe required height of release in terms of R. b) Now assume therelease point is at or above the minimum release height. Show thatthe normal force on the car at the bottom of the loop exceeds thenormal force at the top of the loop by six times the weight of thecar. The normal force on each rider follows the same rule.

Explanation / Answer

Conservation of Energy: Ei = EfKEi + Ui +... = KEf + Uf + ... KE1 + U1 = KE2 + U2 0 + mgh = (1/2)mvf2 + 0 KE2 + U2 = KE3 + U3 (1/2)mv2 + 0 = 0 + mg(2R) E1 = E2 = E3 = mgh = mg(2R) h = 2R F2 = -dE2/dr = -2mg Fbottom = F2 - mg = -2mg - mg = -3mg Ftop = -F2 - mg = 2mg -mg = mg Fbottom = - 3Ftop =-3mg

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