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

The minimum height h for which the roller coaster will not fall off the track at the top of the loop is when the normal force exerted by the track on the roller coaster is zero. This occurs at the top of the loop, where the gravitational force is acting directly downwards and the centripetal force (provided by the normal force) is acting directly upwards.
Using conservation of energy, we can find the velocity of the roller coaster at the top of the loop:
mgh = 1/2 mv^2
where m is the mass of the roller coaster, g is the acceleration due to gravity, and v is the velocity of the roller coaster at the top of the loop. Solving for v, we get:
v = sqrt(2gh)
Now, at the top of the loop, the centripetal force is provided by the normal force:
mv^2/R = N
where N is the normal force. Setting N to zero (since that is the minimum height at which the roller coaster will not fall off the track), we get:
mv^2/R = 0
or
v = 0
Substituting our expression for v, we get:
sqrt(2gh) = 0
or
h = 0
This means that the roller coaster will fall off the track at the top of the loop unless the height is greater than or equal to the radius of the loop. Therefore, the minimum height h for which the roller coaster will not fall off the track at the top of the loop is:
h = R = 6.0 m
and the best choice is C.

A Roller Coaster Starts from Rest at a Height H