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Unformatted text preview: Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU Qualification Exam Day 1 (February 2006) (1) A pendulum of mass m and length b is mounted to a heavy sled of mass m β which is free to slide with negligible friction down an inclined plane of slope α as shown. The entire system acts under the influence of gravity, where g is the downward acceleration due to gravity. (a) Let φ denote the angle that the pendulum bob makes with the vertical and s denote the distance that the sled has slid down the slope (see diagram). Determine the Lagrangian ( ) φ φ & & , , , s s L describing the system in terms of this angle and distance and their time derivatives. The terms to consider are: The kinetic energy of the main body of the sled: 2 2 1 s m K body & β =The kinetic energy of the pendulum bob: ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] 2 2 2 2 cos 2 2 1 sin cos 2 1 φ α φ φ α φ φ α φ φ & & & & & & & b s s b m b s b m K bob + + + = + + + + =The potential energy of the main body of the sled: ( )...
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 Spring '12
 Dr.Jaouni
 Energy, Friction, Kinetic Energy, Mass, Cos, Pφ, bob hangs—this

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