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Unformatted text preview: March 19, 2010 Cornell University, Department of Physics P3318, Analytical Mechanics, HW#8, due: 4/2/10, 2:30PM (at section) Question 1 : Harmonic oscillator and Friction Consider a system with one DOF and friction such that we modify the EOM to be d dt parenleftBigg ∂L ∂ ˙ q parenrightBigg − ∂L ∂q = − ∂f ∂ ˙ q (1) 1. Show that the rate of energy going out of the system is given by dE dt = − ˙ q ∂f ∂ ˙ q (2) Note that we define the energy to be the quantity that is conserved in the f = 0 case. 2. From this point on we consider the case where V = kx 2 / 2 and we expand f and keep the leading term, that is, f = λ ˙ q 2 . Find the EOM for that case. 3. Show that in the specific case above, the system can be described by the following time dependent Lagrangian L = e- 2 λt ( T − V ) . (3) Question 2 : Friction and external force Consider a harmonic system with an external force such that the EOM is ¨ x + 2 λ ˙ x + ω 2 x = F m (4) and F = f exp( αt ) cos( γt...
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- Spring '08
- Friction, parametric resonance