Help? A single-car loop-the-loop ride is launched by means of a giant spring (k

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A single-car loop-the-loop ride is launched by means of a giant spring (k = 30 kN / m). The car (m = 900 kg) is placed at rest against the spring which has been compressed by a distance, d. When released, the spring propels the car into a frictionless vertical circular track of radius R = 7.0 m. Using the methods of Chapter 8, find an algebraic expression (that is without any numerical values for the quantities) for the smallest that d can be if the car if the car is to stay on track. (In section 6-5, your book examines a loop-the-loop bicycle stunt using Newton's 2nd Law. You may want to review this example.) Find a numerical value for d.

Explanation / Answer

a) at top of loop

mg = mv^2/R
v = sqrt(gR)

conservation of energy
Ei = Ef

1/2 k d^2 = 1/2 mv^2 + m g 2 R

1/2 k d^2 = 1/2 m g R + 2 m g R

k d^2 = m gR + 4 m g R = 5 m g R

d = sqrt( 5 m g R/k)

b) d = sqrt(5*900*9.81*7/30.0E3)=3.21 m

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