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ph409_test3ANS - Physics 409/Classical Mechanics Name...

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Physics 409/Classical Mechanics Test #3 19 October 2010 Name ANSWERS Answer all parts of each of the two questions. 25 points apiece. 50 points total. Begin the answer to each question on a new sheet of paper. Clearly define all coordinates and variables. Be sure to include sufficient detail in each answer so that the logic you are using is clear. (1) Consider the two-body conservative central force problem in polar coordinates r and θ . The effective radial equation of motion is . Here, l is the constant angular 2 3 ( / ) ( ) r l r f r μ μ = + ±± momentum, μ is the reduced mass, and f ( r ) is the central force, derivable from a potential V ( r ). For the attractive potential V ( r ) = ! k / r , the equation of the orbit has the form r = c/ (1 + e cos θ ), where c = l 2 / ( μk ), k is a positive constant, and e is a nonnegative constant known as the eccentricity. The energy of any possible orbit with angular momentum l and eccentricity e can be expressed generally as . E k l e = μ 2 1 2 2 ( ) (a)(10 pts.) Add l 2 /(2 μr 2 ) to V ( r ) to obtain the effective one-dimensional potential for this V problem. Draw a sketch of versus r , and indicate E on it for the four cases: (i) e >1, (ii) e= 1, V (iii) 1> e > 0, and (iv) e =0 . Qualitatively describe the type of motion possible for each of these cases. Indicate the allowable ranges for the radial coordinate. It should be possible to have a circular orbit for your effective potential. Is this circular orbit stable or unstable? Explain. ANS: See Figures 3.3, 3.4, 3.6, and 3.8 in Goldstein, either edition. (b)(3 pts.) Find the radius r 0 and period τ 0 for the stable circular orbit. Express τ 0 in terms of r 0 , μ , and k . ANS: From the circular orbit condition, f ef = ! d V N ( r ) /dr = 0, we have l 2 /( μr 0 3 ) ! k / r 0 2 =
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