Dynamics_Part15

Dynamics_Part15 - 3 F z = N − mg = m V 2 B R The...

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Problem Set 2: Problem 4. Problem: The radius of curvature at the bottom and at the top of a roller coaster is R .Th em a s so fa roller-coaster car is m . Express your answers in terms of R and the acceleration of gravity, g . (a) If the contact force between a car and the rail must not exceed six times the weight of the car, what is the maximum speed, V B ,a tthebo t tom? (b) What is the speed of a car at the top, V T , that yields zero contact force? Solution: The following figure shows the forces acting on a car at the bottom and at the top of the roller coaster. Specifically, in the vertical direction, we have the weight of the car, mg , and the contact force, N . R R N mg V B N mg V T •• •• . . ................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......... . . . . . . . . . . . . . . . . ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................... . . .
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Unformatted text preview: 3 F z = N − mg = m V 2 B R The acceleration is positive upwards because it is directed toward the center of rotation, which lies above the roller-coaster track. Thus, the contact force is N = mg + m V 2 B R Clearly, the contact force increases as the speed increases, which tells us that the maximum speed occurs when N = 6 mg , so that 6 mg = mg + m V 2 B R = ⇒ V 2 B = 5 g R Thus, the maximum speed is V B = 5 g R (b) At the top of the roller coaster, Newton’s Second Law tells us that in the vertical direction 3 F z = N − mg = − m V 2 T R The acceleration is positive downwards because it is directed toward the center of rotation, which lies below the roller-coaster track. Thus, the contact force is N = mg − m V 2 T R...
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