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Final-SP07 - Phys 208 Theoretical Physics Final exam 8i...

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Phys 208 – Theoretical Physics – Final exam – May 10, 2007 1.(15 pts) a) Find and plot the roots of 3 8 i . b) Evaluate ( ) i i in Cartesian form, i.e. , x iy + form. c) Determine the points in the ( x, y ) plane that satisfy the equation | 2 3 | 4 z i + = . 2.(20 pts) Find the exponential Fourier transform of the given function ( ) f x and write ( ) f x as a Fourier integral [that is, find ( ) g α and substitute your result into the integral for ( ) f x ]. 1, 1 0 ( ) 1, 0 1 0, | | 1 x f x x x < < = < < > a) Show that you obtain 0 cos 1 sin( ) ( ) 2 x d f x α π α α α = and list the possible results of the integral that you can obtain depending on the value of x . b) Using Parseval’s theorem for Fourier transforms , what would be the result for the integral given by 2 2 0 (cos 1) d α α α Recall Parseval’s theorem is 2 2 1 | ( ) | | ( ) | 2 g d f x dx α α π −∞ −∞ = 3.(15 pts) A ball of mass M and radius R rolls without slipping down an inclined plane under the action of gravity.
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