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Physics 325 Spring 2011 Problem Session 6 Solutions

# Physics 325 Spring 2011 Problem Session 6 Solutions -...

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Physics 325 Spring 2011 Discussion 6 February 28, 2011 A particle of mass μ and initial angular momentum is moving in a central force field F ( r ) = kr, with k > 0 . Starting from the central force equation u 00 + u = - μ 2 u 2 F 1 u , where u 0 = du/dθ and u = 1 /r , we want to solve for the particle’s path using the following steps. (a) If u 0 = 0 at u = 1 /s , solve for ( u 0 ) 2 as a function of u . Here, s is given by s = 2 μk 1 / 4 (b) Assuming u 0 0 , integrate once more to solve for u as a function of θ . Use initial condition r 2 sin α = s 2 at θ = 0 , for a given α . Hint: Make the substitution u 2 = a 2 sin φ , where a is some constant chosen appro- priately. (c) Using x = r cos θ and y = r sin θ , show that the previous result can be rewritten as s 2 = ( x 2 - y 2 ) sin α + 2 xy cos α. (d) Let σ 2 + λ 2 = 1 and sin α = σ 2 - λ 2 , cos α = 2 σλ. (We need σ 2 + λ 2 = 1 so that sin 2 α + cos 2 α = 1 .) Show that part (c)’s result can also be written as s 2 = ˜ x 2 - ˜ y 2 , where ˜ x = σx + λy, ˜ y = λx - σy.

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Physics 325 Spring 2011 Problem Session 6 Solutions -...

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