chem5314_hwk4 - Chem 5314 homework #4 – due Nov. 3, 2011...

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Unformatted text preview: Chem 5314 homework #4 – due Nov. 3, 2011 Problem 1 – reduced mass a) Consider two particles of masses m 1 and m 2 in one dimension, interacting through a potential that depends only upon their relative separation ( x 1- x 2 ), so that V ( x 1 ,x 2 ) = V ( x 1- x 2 ). Given that the force acting upon the j th particle is f j =- ( ∂V/∂x j ), show that f 1 =- f 2 . What law is this? b) Newton’s equations for the two particles are m 1 d 2 x 1 dt 2 =- ∂V ∂x 1 and m 2 d 2 x 2 dt 2 =- ∂V ∂x 2 (1) Now introduce center of mass and relative coordinates by X ≡ m 1 x 1 + m 2 x 2 M x ≡ x 1- x 2 (2) where M = m 1 + m 2 , and solve for x 1 and x 2 to obtain x 1 = X + m 2 M x and x 2 = X- m 1 M x (3) Show that Newton’s equations in these coordinates are m 1 d 2 X dt 2 + m 1 m 2 M d 2 x dt 2 =- ∂V ∂x (4) and m 2 d 2 X dt 2- m 1 m 2 M d 2 x dt 2 = + ∂V ∂x (5) c) Now add these two equations to find M d 2 X dt 2 = 0 (6) Interpret this result....
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This note was uploaded on 02/08/2012 for the course CHEM 5314 taught by Professor Nielsen during the Fall '11 term at University of Texas at Dallas, Richardson.

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chem5314_hwk4 - Chem 5314 homework #4 – due Nov. 3, 2011...

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