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Unformatted text preview: Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 2006 Quiz 1 October 4, 2006 Instructions • Do not start until you are told to do so. • Solve all problems. • Put your name on the covers of all notebooks you are using. • Show all work neatly in the blue book, label the problem you are working on. • Mark the final answers. • Books and notes are not to be used. Calculators are unnecessary. 2 Useful Formulae Newton and Basic Kinematics: F = p ˙ = ma for v c t = t v = v + dt a t =0 t = t = t r = r + v t + dt t dt F ( t ) /m t =0 t =0 Lagrangian and Hamiltonian: L ( q, q ˙) = T − U ; H ( p, q ) = T + U = pq ˙ − L Euler-Lagrange (without and with constraints): ∂L d ∂L ∂L d ∂L ∂g a − = 0; − + λ a = 0 ∂x dt ∂x ˙ ∂x dt ∂x ˙ ∂x a Polar Coordinates: x = r sin θ cos φ ; y = r sin θ sin φ ; z = r cos θ Cylindrical Coordinates: x = r cos φ ; y = r sin φ ; z = z Possibly useful integrals: dx dx =...
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- Spring '08
- Mass, Variable length, Polar coordinate system, Double pendulum, angular velo city, Variable Length Pendulum