GOAL Combine centripetal force with conservation of energy. Derive the results s

Question

GOAL Combine centripetal force with conservation of energy. Derive the results symbolically. PROBLEM Figure (a) shows a roller coaster car moving around a circular loop of radius R. (a) What speed vbot must the car have at the top of the loop so that it will just make it over the top without any assistance from the track? (b) What speed will the car subsequently have at the bottom of the loop? (c) What will be the normal force on a passenger at the bottom of (a) A roller coaster traveling around a nearly circular track. Kb) A jet executing a vertical loop. the loop if the loop has a radius of 10.0 m? STRATEGY This problem requires Newton's second law and centripetal acceleration to find an expression for the car's speed at the top of the loop, followed by conservation of energy to find its speed at the bottom. If the car just makes it over the top, the force n must become zero there, so the only force exerted on the car at that point is the force of gravity, mg. At the bottom of the loop, the normal force acts up toward the center and the gravity force acts down, away from the center. The difference of these two is the centripetal force. The normal force can then be calculated from Newton's second law.

Explanation / Answer

at top

N+mg = m*ac

ac = centripetal acceleration

ac = v^2/r = (1.5 x 10^2)^2/8*10^2

ac = 28.125 m/s^2

N = m(ac-g)

N = 70 * (28.125 - 9.8)

N = 1282.75 N

part b )

at bottom

N-mg = mac

N = m(g+ac)

N = 2654.75 N

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