A roller-coaster car may be represented by a block of mass 50.0 . The car is rel

Question

A roller-coaster car may be represented by a block of mass 50.0 . The car is released from rest at a height = 45.0 above the ground and slides along a frictionless track. The car encounters a loop of radius = 15.0 at ground level, as shown. As you will learn in the course of this problem, the initial height 45.0 is great enough so that the car never loses contact with the track.

a) Find an expression for the kinetic energy of the car at the top of the loop.
b) Find the minimum initial height at which the car can be released that still allows the car to stay in contact with the track at the top of the loop.

Explanation / Answer

a) Find an expression for the kinetic energy of the car at the top of the loop. KE (loop) = PE (released point) - PE ( top of the loop) = m g h - m g 2R = m g ( h - 2R) b) There mg = mv^2 / R = 2 mg (h -2 R) / R ==> m g R = 2 m gh - 4mgR ==> 5 R = 2 h ==> h = 2.5 R Substitute values.

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