Question

Question 13 (1 points) Consider a roller coaster with a single, circular loop of radius R. Assuming no friction and no external driving once the car reaches the top of the first hill, at what height must the car start in order to make it around the loop? Assume that the car wheels cannot grip the rails, so that if the normal force at the top drops to zero then the car will fall off the loop. Give your answer as a multiple of R. Image size: SML Max Please enter a numerical answer below. Accepted formats are numbers or "e" based scientific notation e.g. 0.23, -2, 106, 5.23e-8 Enter answer here

Answer 1

Suppose mass of the car = m

Suppose, v is the speed of the car at top of the loop.

Centrifugal force at this point, Fc = m*v^2 / R

Weight of the car, W = m*g

The car will not loose the contact with the loop, when -

Fc > W

In the limiting case -

Fc = W

=> m*v^2 / R = m*g

=> v^2 = (m*g*R) / m = g*R ---------------------------------------------(i)

Initial height of the car = h = Yi

Loss in potential energy at the top of the loop = m*g*(Yi - 2R)

Gain in kinetic energy by the car = (1/2)*m*v^2

By conservation of energy -

loss in potential energy = gain in kinetic energy

=> m*g*(Yi - 2R) = (1/2)*m*v^2

=> 2g*(Yi - 2R) = v^2

put the value of v^2 from equation (i) -

2g*(Yi - 2R) = g*R

=> 2*(Yi - 2R) = R

=> 2*Yi - 4R = R

=> 2*Yi = 5R

=> Yi = (5/2)R = 2.5*R (Answer)