Centripetal Acceleration at top of Loop

[Sylvia]

Homework Statement

A roller coaster is on a track that forms a circular loop in the vertical plane. If the car is to just maintain contact at the top of the loop, what is the minimum value for its centripetal acceleration at this point?
A) 2g downward
B) g downward
C) 2g upward
D) g upward
E) 0.5 downward

Homework Equations

fcp = m(v^2 / r)

The Attempt at a Solution

I thought since normal force and weight are pointing in the same direction, the Fcp would be mg + mg = 2mg. But the correct answer is g downward. Please explain.

 

[mfb]

In the case discussed here, there is no normal force. Centripetal acceleration is purely coming from gravity.

 

[Sylvia]

I don't understand why there's no normal force. It says it "just maintains contact" which means the cart and rail are in contact, meaning that there should be a normal force, right? Thank you for replying.

 

[mfb]

They are "just in contact" - they are directly at the border between "separating" and "contact with a contact force" - therefore, zero force but still some sort of contact.

 

[HalfLostAndSearching]

The correct answer is g downward because at the top of the loop, the normal force and weight are in opposite directions, not the same direction. This is because the normal force is always perpendicular to the surface of contact and at the top of the loop, the surface of contact is horizontal, while weight is acting downward due to gravity. Therefore, the net force acting on the car at the top of the loop is only the weight, which is equal to mg.

The equation for centripetal force, fcp = m(v^2 / r), tells us that the magnitude of the centripetal force is equal to the mass of the object (m) multiplied by the square of its velocity (v^2) divided by the radius of the circle (r). In this case, the velocity of the car is constant and the radius of the loop is also constant. Therefore, the only way to change the magnitude of the centripetal force is by changing the mass of the car.

In the case of the roller coaster, the car needs to maintain contact with the track at the top of the loop, which means that the normal force must be equal to or greater than zero. The normal force is provided by the track and is equal to the centripetal force needed to keep the car moving in a circular motion. Since the car is at the top of the loop, the normal force must be equal to or greater than the weight of the car, mg. Therefore, the minimum value for centripetal acceleration at the top of the loop is g downward, which is equal to the weight of the car divided by its mass (mg/m = g).

In conclusion, the correct answer is g downward because at the top of the loop, the only force acting on the car is its weight, which is equal to the minimum value for centripetal force needed to maintain contact with the track. This minimum value is g downward because the normal force must be equal to or greater than the weight of the car in order to maintain contact.