homework7sol.20100310.4b986e6d5779f9.40121625

homework7sol.20100310.4b986e6d5779f9.40121625 - of mass...

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TAM 412, Homework 7, Selected answers Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign danko@illinois.edu 1. Here T = m 2 ¡ _ x 2 + _ y 2 + _ z 2 ¢ and U = mgz where x = r cos μ;y = r sin μ;z = f ( r ) It follows that the angular momentum about the Z -axis given by ¯ ¯ ¯ ¯ ¯ ¯ 0 0 1 x y z m _ x m _ y m _ z ¯ ¯ ¯ ¯ ¯ ¯ is a conserved quantity. 2. Let u r = 0 and u μ = 1 ) r = ~ r and μ = ~ μ + s . It follows that u x = ¡ r sin μ = ¡ y , u y = r cos μ = x , and u z = 0, i.e., a rotation about the Z -axis. It follows that d ds L ³ ~ r; _ ~ r; ~ μ + s; _ ~ μ ´ ¯ ¯ ¯ ¯ s =0 = 0 i.e., u r @L @ _ r + u μ @L @ _ μ = r 2 _ μ is conserved. 6. Here T = m 1 2 ¡ _ x 2 1 + _ y 2 1 + _ z 2 1 ¢ + m 2 2 ¡ _ x 2 2 + _ y 2 2 + _ z 2 2 ¢ and U = f μq ( x 2 ¡ x 1 ) 2 + ( y 2 ¡ y 1 ) 2 + ( z 2 ¡ z 1 ) 2 where the cartesian coordinates are relative to an inertial reference frame with origin at the center
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Unformatted text preview: of mass (this is inertial because of the result of exercise 3.7). Consider the generators u x 1 = y 1 , u y 1 = x 1 , u z 1 = 0, u x 2 = y 2 , u y 2 = x 2 , and u z 2 = 0, such that x 1 = ~ x 1 cos s ~ y 1 sin s y 1 = ~ x 1 sin s + ~ y 1 cos s z 1 = ~ z 1 x 2 = ~ x 2 cos s ~ y 2 sin s y 2 = ~ x 2 sin s + ~ y 2 cos s z 2 = ~ z 2 and conclude that the total angular momentum about the Z-axis y 1 _ x 1 + x 1 _ y 1 y 2 _ x 2 + x 2 _ y 2 is a conserved quantity. 1...
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This note was uploaded on 11/17/2011 for the course TAM 412 taught by Professor Weaver during the Spring '08 term at University of Illinois, Urbana Champaign.

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