Question

A roller coaster car with its passengers has a mass of 220 kg and it goes through a series of horizontal and vertical curves.

A. Draw the free body diagrams for the car going around a horizontal curve, the bottom of a vertical loop, and the top of a vertical loop in which the cart is upside down. Indicate the direction of the acceleration on each diagram.

B. If the car is moving at 25 m/s what is the smallest horizontal curve it can go around and not exceed a 4.5g limit on acceleration (g = 9.8 m/s2?

C. What speed does the car need at the top of a loop with a radius of 4.5 m in order for the car to experience weightlessness?

D. What is the apparent weight of the car at the bottom of a 4.5 m radius loop at 25 m/s?

Answer 1

A. When the car goes around a horizontal curve, the Normal force and the Gravity force (Weight) are perpendicular to each other at all times, and the Weight never acts as a centripetal force (it never points to the center of the circular path). On the other hand when the car is going through a vertical loop the situation is different, because the line of action of mg is in the radial direction. When the cart is on top, mg is alligned with the Normal force, when the cart is at the bottom, mg is opposite to the Normal:

B. For any circular motion the centripetal acceleration and the velocity at a point in the circumference are related by:

If we want the car not to exceed the 4.5g limit the minimum radius should be:

C. When the cart is at the top of a vertical loop the balance in the radial direction is:

The sensation of weightlessness appears when N=0. In this case:

D. The apparent weight of the cart at the bottom is given by the Normal force. The balance in the radial direction in this case is:

This is about 17 times the real weight of the car