P1. Roller Coaster Loop The figure below shows a roller coaster of mass m =500 k

Question

P1. Roller Coaster Loop The figure below shows a roller coaster of mass m =500 kg that can roll down a frictionless track and enter a vertical loop of radius R = 20 m. Ignore air resistance.

A. What is the minimum speed required at the top (Point B) so that the roller coaster does not fall off the loop at point B? (Ignore friction and air resistance.)

B. Find the minimum required starting height h at point A to ensure that the roller coaster does not fall off the loop at point B? (Ignore friction and air resistance.)

C.Point C is half way through the loop (i.e. Point C is 20m above the ground). What is the magnitude of the centripetal acceleration experienced by riders in the roller coaster at point C, if it started at rest from point A? Express your answer in terms of the freefall acceleration, g.

Explanation / Answer

A)
In case of minimum speed the normal force at point B will be zero and the force due to gravity will provide the centripetal acceleration. So,
mg = mv^2/R
so v_min = sqrt(Rg) = sqrt(20*9.81) = 14 m/s
B)
To satisfy this condition the potential energy difference will b equal to the kinetic energy at point B.
mg(h-2R) = 0.5mv^2
g(h-2R) = Rg/2
h-2R = 0.5R
h = 2.5 R = 50 m
C)
Work done by gravity force from point B to point C = mgR hence change in kinetic energy = mgR
mgR = 0.5m(v^2-Rg)
Rg = 0.5(v^2-Rg)
v = sqrt(3Rg) = sqrt(3*20*9.81) = 24.26 m/s
hence centripetal acceleration = v^2/R = 3g = 29.43 m/s^2

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