ph409_test6ANS - Physics 409/Classical Mechanics Test#6 7...

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Physics 409/Classical Mechanics Test #6 7 December 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. (/ ) ii i hq L q L =∂ ±± (1) (25 pts.) A simple pendulum (mass m at the end of massless rod of length l ) is attached to a cart constrained to move at constant speed v 0 in the x direction. The pendulum oscillates in the x - y plane only, where y is the vertical direction (the direction of gravity). Assume y = 0 where the pendulum attaches to the horizontal support bar. (a) (13 pts.) Using appropriate coordinates set up a correct Hamiltonian for the system. (The Lagrangian and energy function h should be explicitly derived.) (b) (4 pts.) Ignoring any additive constant terms, what physical quantity does your Hamiltonian represent? Is your Hamiltonian conserved? (c) (4 pts.) Find the Hamiltonian equations of motion for the system. (d) (4 pts.) Combine the Hamiltonian equations of motion and show that they reduce to a single EOM for a simple pendulum. ANS: Let x = v 0 t + l sin θ and y = ! l cos θ . The Lagrangian is 22 00 ( / 2)[( ) 2 v v ] Lm l l θθ =+ + and with we find cos mgl θ + 0 /( v c o s ) pL m l l ∂ = + 0 ( / 2)[( ) v ] hm l =− ± .
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ph409_test6ANS - Physics 409/Classical Mechanics Test#6 7...

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