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Unformatted text preview: Problem: Orbits with a 6-4 central potential (a) The equation of motion for the one-dimensional radial problem is r = f ef ( r ) = dV dr + l 2 r 3 , (1) with the 6-4 potential is de f ned as V ( r ) = k a r 6 a r 4 , (2) where k is a constant that determines the strength of the potential, a is a constant with dimensions of length, l is the constant angular momentum, r 2 , and is the reduced mass of the two particle system. This equation can be written explicitly as m r = f ef ( r ) = k a 6 a r 7 4 a r 5 + l 2 r 3 = dV dr , (3) where V is the e f ective potential for this problem, V = V ( r ) + l 2 2 r 2 . (4) To discuss the qualitative nature of the motion, we need to make a plot of V versus r . It will be convenient to introduce a scaled radial coordinate x = r/a . This allows us to write e V ( x ) = V k = c 2 x 2 + x 6 x 4 , (5) where c is the positive constant c = l 2 ka 2 . (6) We can make convenient choices for the value of c by investigating when circular orbits are possible. The condition to be satis f ed is that the e f ective force should vanish, f ef ( r c ) = k a " 6 a r c 7 4 a r c 5 # + l 2 r 3 c = 0 , (7) where r c is the radius of the circular orbit. Using Eq.(6) it follows that 6 x 7 c 4 x 5 c + cx 3 c = 0 , (8) where x c = r c /a . After multiplying by x 7 c , Eq.(8) becomes a quadratic equation, Eq....
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