Question

In the homework, you were asked to calculate the minimum height needed for a roller coaster car to just barely make a circular loop-the-loop. Instead, suppose a roller coaster designer wants the riders to feel their seats pushing back on them with a force equal to their weight when the car is at the inside, top of the loop. What height h should the car be released from if this condition is to be met? Express your answer in terms of R. Ignore friction.  
a. After drawing a free body diagram for a person in the car when the car is at the inside, top of the loop, determine a constraint equation on the square of the speed when the car is at that position. b. Apply conservation of energy to determine the necessary, minimum height h. Express your answer in terms of the radius R of the loop.

Answer 1

At a top of the loop, we have

F = F_N + (m g) = (m v² / R)

F_N = (m v² / R) - (m g)

Where, F_N = normal force = 0

0 = (m v² / R) - (m g)

(m v² / R) = (m g)

v² = g R                                                                        { eq.1 }

What height 'h' should the car be released from if this condition is to be met?

Using conservation of energy, we have

K.E_initial + P.E_initial = K.E_final + P.E_final

0 + m g h = (1/2) m v² + m g (2R)                                           { eq.2 }

Inserting the value of 'v' in eq.2 & we get

m g h = (1/2) m g R + m g (2R)

m g h = (5/2) m g R

h = (5/2) R

Answer in terms of 'R'.