Modern roller coasters have vertical loops like the one shown in the figure. The

Question

Modern roller coasters have vertical loops like the one shown in the figure. The radius of curvature is smaller at the top than on the sides so that that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into thier seats.

A. The beginning of this roller coaster is at the top of a high hill. If it started from rest at the top of this hill, how high, in meters, above the top of the loop is this initial starting point? You may assume there is no friction anywhere on the track.

B. If it actually starts 3.5 m higher than your answer to the previous part (yet still reaches the top of the loop with the same velocity), how much energy, in joules, did it lose to friction? Its mass is 1600 kg.

Explanation / Answer

at the top of loop.

mg - N = m a_C = m v^2 / r_min

it will loose contact if N becomes zero.
for limiting case (minimum velocity at top)

N = 0

m g = m v^2 / r_min

v= sqrt(r_min g )

now applying energy conservation,

m g h + 0 = m g (2r) + m v^2 / 2

m g h = 2 m g r + m g r/2

h = 5 r / 2 or 2.5 r

(B) now H = h + 3.5

m g H = 2 m g r + m v^2 /2

2.5 m g r + 3 m g = 2 m g r + m v^2 /2

v^2 /2 = 5 m g r / 2 + 6 m g / 2

v = sqrt(5 m g r + 6 m g )

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