Hw2_prob3_gps21_g19

HW2_prob3_GPS21_G19
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Goldstein 1-19 (3 rd ed. 1.21) Start with a Cartesian coordinate system with the origin at the hole. In these coordinates, the kinetic energy is T = 1 2 m 1 ( · x 2 + · y 2 )+ 1 2 m 2 · z 2 , (1) where m 1 and m 2 are the particle masses, x and y determine the position of particle 1 in the plane, and z determines the position of particle 2 below the hole. The system has only two degrees of freedom because the position of particle 2 on the z axis is constrained by the string length s . Use the polar coordinates r and θ as the two generalized coordinates for this system, x = r cos θ , (2) y = r sin θ , (3) and the constraint is z = r s. (4) In terms of r and θ , we have the following expression for T T = 1 2 ( m 1 + m 2 ) · r 2 + 1 2 m 1 ( r · θ ) 2 , (5) and the potential energy is V = m 2 gz = m 2 g ( r s ) . (6) It follows that the Lagrangian, L = T V ,is L = 1 2 m · r 2 + 1 2 m 1 ( r · θ ) 2 m 2 g ( r s ) , (7) where m is the total mass m = m 1 + m 2 . (8) 1
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