A roller coaster starts from rest at a height h at the left side of a loop-the-loop, as shown in the figure. It is not attached to the track in anyway, and there is no friction from the track or from air resistance. If the radius of the loop is R = 6.0 m, what is the minimum height h for which the roller coaster will not fall off the track at the top of the loop?
A) 21 m
B) 18 m
C) 15 m
D) 12 m
E) 8.5 m

Question

A roller coaster starts from rest at a height h at the left side of a loop-the-loop, as shown in the figure. It is not attached to the track in anyway, and there is no friction from the track or from air resistance. If the radius of the loop is R = 6.0 m, what is the minimum height h for which the roller coaster will not fall off the track at the top of the loop? 
A) 21 m 
B) 18 m 
C) 15 m 
D) 12 m 
E) 8.5 m

Quizplus · Accepted Answer

At the top of the loop, the roller coaster experiences an apparent weight of zero, meaning that the only force acting on it is the force of gravity. The minimum height needed for the roller coaster to stay on the track at the top of the loop occurs when the normal force provided by the track is equal to zero. This occurs when the centrifugal force (mv^2/R) is equal to the force of gravity (mg), so:

mv^2/R = mg

v^2 = gR

v = sqrt(gR)

At the top of the loop, the roller coaster must be moving at this speed in order to stay on the track. The roller coaster reaches this speed at the bottom of the loop, so we can use conservation of energy to find the minimum height needed for the roller coaster to reach this speed:

mgh = (1/2)mv^2

h = (1/2)(v^2/g)

h = (1/2)((gR)/g) = R/2 = 3.0 m

Therefore, the minimum height needed for the roller coaster to stay on the track at the top of the loop is h = R/2 = 3.0 m, which corresponds to choice C.

A Roller Coaster Starts from Rest at a Height H