hw4 - Physics 318 Problem Set 4 Due Wednesday 1 A particle...

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Physics 318: Problem Set 4 Due Wednesday, February 20, 2008 1. A particle is constrained to move without friction along the curve y = y ( x ) in two dimensions in a uniform gravitational field. It starts from rest at ( x 1 , y 1 ) and ends at ( x 2 , y 2 ), where y 1 = y ( x 1 ) and y 2 = y ( x 2 ). a. Show that the time taken to reach the second point is T [ y ] = integraldisplay x 2 x 1 dx radicalbig 1 + y ( x ) 2 radicalbig 2 g [ y 1 y ( x )] . b. Compute the choice of function y ( x ) which minimizes T [ y ] as follows. Use the fact that the integrand in T [ y ] does not depend on x to obtain a first integral of the Euler equation (ie to find a relation between y and y ). Integrate this differential equation using a substitution of the form y = k 2 sin 2 ( ϕ/ 2) h with suitable constants k and h . Show that the resulting curve is a cycloid [a curve given parametrically by x x 1 = a ( ϕ sin ϕ ) and y 1 y = a (1 cos ϕ ) for some constant a ] with a cusp at the point ( x 1 , y 1 ). 2. Generalize the derivation of the Euler equation for the function y ( x ) which minimizes the functional J [ y ] = integraltext x 2 x 1 dxF ( y, y , x
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