Find an expression for the kinetic energy ofthe car at the top of the loop. A ro

Question

Find an expression for the kinetic energy ofthe car at the top of the loop. A roller coaster car may be approximated by a block of mass m. The car, which starts from rest, isreleased at a height h above the ground and slides along africtionless track. The car encounters a loop of radius R, as shown. Assume that the initial heighth is great enough so that the car neverloses contact with the track. Find an expression for the kinetic energy ofthe car at the top of the loop. Express the kinetic energy in termsof m, g, h, and R. K =? Find the minimum initial height h at which the car can be released that stillallows the car to stay in contact with the track at the top of theloop. Express the minimum height in termsof R. hmin =?

Explanation / Answer

kinetic energy = loss of potential energy .     K = m g (h - 2R ) . Now for the minimum height... first use the idea above: .     (1/2) m v2 = m g (h- 2R) . And the idea that the car is barely in contact at the topmeans that gravity provides the centripetal force; .          mg = mv2 /R         or            mv2 = mgR . So now... .      (1/2) mgR = mgh - 2mgR . Simplify: .             R   =   2h   - 4R .         h = 5R /2

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