Consider a roller coaster that is initially at rest at (1) Under the influence o

Question

Consider a roller coaster that is initially at rest at (1) Under the influence of gravity, the roller coaster car glides frictionlessly down the rail where it then enters a vertical circular loop of radius R at (2). Continuing in circular motion, the car travels through the INSIDE of the loop to pass through (3), (4), and then (5) to reach position (6). Assume R is much larger than the physical dimensions of the roller coaster car.

a) Determine the minimum height (either h1 or hmin) from which the roller coaster car must be released at (1) in order to barely reach position (4) without losing contact with the rails.
b) Suppose you are one of the passengers aboard this roller coaster. If h1 = hmin, determine the g-force you would experience immediately upon entering the vertical loop at (2). [ g-force = FN/mg ]
c) Determine the work done by gravitational force on a passenger (of mass m) as the car is displaced from (2) to (4) along the semicircular loop.

Explanation / Answer

a) TO barely reach position 4

mv^2/R = mg

v^2 = gR

v = sqrt(gR)

Conservation of energy

TE1 = TE4

PE1 + KE1 = PE4 + KE4

mgh1 + 0 = mgh4 + 0.5 mv2

mgh1 = 2mgR + 0.5 m (gR)

mgh1 = 2.5 m g R

h1 = (2.5 R)

h

min= 2.5 R

b) F N = mg + mv^2/r

= mg + m(2gh1)/R

= mg + m *2*g*2.5 R / R

   = 6 mg

G force = FN / mg = 6

c) Work done by gravity force = - mg (2R)

   = - 2mgR

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