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Unformatted text preview: 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 )....
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