PHYS 598 hw8 - 1 Pantograph Drag A high-speed train picks...

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Physics 498/MMA Handout 8 Fall 2007 Mathematical Methods in Physics I m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Pantograph Drag : A high-speed train picks up its electrical power via a pantograph from an overhead line. The locomotive travels at speed U and the pantograph exerts a constant vertical force F on the power line. F U Rheil Cymru A high-speed train. We make the usual small amplitude approximations and assume (not unrealistically) that the line is supported in such a way that its vertical displacement obeys an inhomogeneous Klein-Gordon equation ρ ¨ y - Ty 00 + ρ Ω 2 y = ( x - Ut ) , with c = p T/ρ , the velocity of propagation of short-wavelength transverse waves on the overhead cable. a) Assume that U < c and solve for the steady state displacement of the cable about the pickup point. (Hint: the disturbance is time-independent when viewed from the train.) b) Now assume that U > c . Again find an expression for the displacement of the cable.
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