ph409_test1ANS - Physics 409/Classical Mechanics Name...

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Physics 409/Classical Mechanics Test #1 16 September 2010 Name ANSWERS Answer all parts of each of the two questions. 50 points total. The points value of each part is indicated. Begin the answer to each question on a new sheet of paper. Clearly define all coordinates and variables. Be sure to include sufficient detail in each answer so that your logic is clear to me. Useful reminders: One form of Lagrange’s equations of motion is −= =• i r F ± ji i jj j dT T Q dt q q q ∂∂ Another form is where L = T - V 0 ± dL L dt q q (1)(25 pts. ) A particle of mass m slides without friction on a very long, rigid, massless wire that is free to rotate about the origin in the x-y plane. No forces act on the particle. (a)( 5 pts. ) Write the kinetic energy T of this system using plane, polar coordinates as generalized coordinates. ( x = r cos θ , y = r sin θ ) 22 2 (/ 2 ) ( ) Tm rr θ =+ ± ± (b)( 8 pts. ) Find the equations of motion (EOM) for this mechanical system. 2 0, ± ±± 20 θθ += ± ± (c)( 5 pts. ) Find an explicit, but general expression for d T /d t . Use the EOM to simplify this expression. Is the kinetic energy T of the particle conserved? /( ) dT dt m rr r + ± ± ±±± ± ([ 2 ] ) 0 mr r r r r r = ++ = ± T is conserved. (d)( 7 pts. ) Find two first integrals for these EOM, and identify two constants of the motion. (Use subscript 0 to designate the initial values of the generalized coordinates and velocities.)
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.

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ph409_test1ANS - Physics 409/Classical Mechanics Name...

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