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hw2 - Physics 318 Problem Set 2 Due Wednesday Feb 6 2008 1...

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Physics 318: Problem Set 2 Due Wednesday, Feb 6, 2008 1. Consider a system of N particles with masses m n , positions r n ( t ), and velocities ˙ r n ( t ). The total mass M and center of mass R ( t ) of the system are defined by M = N summationdisplay n =1 m n , R ( t ) = 1 M N summationdisplay n =1 m n r n ( t ) . We define r n ( t ) = r n ( t ) - R ( t ) to be the position of the n th particle relative to the center of mass. a. Show that the kinetic energy T of the system can be written as the kinetic energy of a point particle with mass M and position R ( t ), plus the kinetic energy of the system relative to the center of mass, that is T = 1 2 M ˙ R ( t ) 2 + 1 2 N summationdisplay n =1 m n ˙ r n ( t ) 2 . (1) b. Show that the angular momentum L of the system can be written as the angular momentum of a point particle with mass M and position R ( t ), plus the angular momentum of the system relative to the center of mass, that is L = M R × ˙ R + N summationdisplay n =1 m n r n × ˙ r n . (2) c. Show that if the system is isolated (so that the external forces
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