s7 - Phys 263/Amath 261 Assignment 6 SOLUTIONS 1. (a) To...

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Unformatted text preview: Phys 263/Amath 261 Assignment 6 SOLUTIONS 1. (a) To set up this problem, use the figure, where the mass m is a distance r from the center of the ring. Then the potential is given by d Φ =- G dM b =- Gaρ b dφ (1) where b is the distance between dM (on the ring) and m (the dotted green line in the figure), and ρ = M/ 2 πa . Let r and r be the position vectors to dM and m , as illustrated: b = | r- r | = | a cos φ ˆ x + a sin φ ˆ y- r ˆ x | (2) = ( a cos φ- r )ˆ x + a sin φ ˆ y | (3) = [( a cos φ- r ) 2 + a 2 sin 2 φ ] 1 / 2 (4) = ( a 2 + r 2- 2 ar cos φ ) 1 / 2 = a " 1 + r a 2 + 2 r a cos φ # 1 / 2 (5) Integrating gives Φ( r ) =- ρaG Z dφ b (6) =- ρG Z 2 π dφ q 1 + ( r a ) 2- 2 r a cos φ (7) as required. (b) We are asked to expand the denominator: " 1 + r a 2 + 2 r a cos φ # 1 / 2 = 1- 1 2 " r a 2 + 2 r a cos φ # (8) + 3 8 " r a 2 + 2 r a cos φ # + · · · (9) = 1 + r a cos φ + 1 2 r a 2 (3 cos 2 φ- 1) + · · · (c) Now we can integrate the expression: Φ(...
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This note was uploaded on 09/20/2010 for the course AMATH 261 taught by Professor Rogermelko during the Spring '10 term at Waterloo.

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s7 - Phys 263/Amath 261 Assignment 6 SOLUTIONS 1. (a) To...

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