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Qu2_solution - Physics 325 Quiz#2 Solutions Wednesday 25...

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Physics 325 Quiz #2 - Solutions Wednesday 25 October 2006 There are 30 points for each correct question for a total of 60 points. 1. A particle of mass m moves in two dimensions under the following potential energy function: U ( r ) = k ( x 2 + 4 y 2 ) / 2 a) Find the force F ( r ). The force is given by F ( r ) = - V , so we have F ( r ) = - V = - ∂x V ( r ) ˆ i - - ∂y V ( r ) ˆ j = - kx ˆ i - 4 ky ˆ j b) Calculate the work done in moving the mass from the point (0 , 0) to the point ( a, b ) using the following two paths: b.i) from x = 0 to x = a along y = 0, followed by y = 0 to y = b along x = a . Call point (0 , 0) = A and call point ( a, b ) = B , then we have W AB = A F ( r ) · dr with dr = dx ˆ i + dy ˆ j For the first path, we have W AB = a 0 - kxdx + b 0 - 4 kydy = - k x 2 2 | a 0 - 2 ky 2 | b 0 = - ka 2 2 - 2 kb 2 b.ii) from (0 , 0) to ( a, b ) along a direct line between these two points. For this path, we need to introduce parametric equations. Let x = at and y = bt , so we have dx = adt and dy = bdt and we are integrating from t = 0 to t = 1: W AB = ( a,b ) (0 , 0) - kxdx - 4 kydy = t =1 t =0 - ka 2 tdt - 4 kb 2 tdt = - ka 2 2 - 2 kb 2 So the work done in moving a mass along these two paths is the same. This is insufficient to prove that the force is conservative, though.
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c) Determine mathematically if the force is conservative.
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