HW3_prob3_GPS14_G10

# HW3_prob3_GPS14_G10 - Goldstein 1.10(3rd ed 1.14 This is a...

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Goldstein 1.10 (3 rd ed. 1.14) This is a two particle system, and the kinetic energy is simply T = m 2 ( v 2 1 + v 2 2 ) . (1) Now, we introduce the COM velocity v and the relative velocities v 0 1 and v 0 2 , v i = v + v 0 i , (2) to obtain, T = mv 2 + m 2 [( v 0 1 ) 2 +( v 0 2 ) 2 ] . (3) In general, we would need six coordinates to describe the con f guration of this system, but because there are three constraints, only three degrees of freedom remain. Thus, we need only three generalized coordinates. If we choose a Cartesian coordinate system with origin at the center of the circle of radius a and we let the circle lie in the x y plane, then the COM location is fully speci f ed by, say, the angular displacement ψ , which we will measure from the x axis. The angle ψ will be our f rst generalized coordinate. In plane polar coordinates, the X and Y components of the center of mass vector R are X = a cos ψ ,Y = a sin ψ . (4) Thus, we have v 2 = · X 2 + · Y 2 = a 2 · ψ 2 . (5) The two remaining degrees of freedom are used to specify the orientation

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HW3_prob3_GPS14_G10 - Goldstein 1.10(3rd ed 1.14 This is a...

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