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HW5_prob_3_GPS13_G11

# HW5_prob_3_GPS13_G11 - Goldstein Problem 2.11(3rd ed 13...

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Goldstein Problem 2.11 (3 rd ed. # 13) While the motion of the mass point is constrained to follow the hoop, the system has only one degree of freedom. To calculate the reaction of the hoop on the particle, we treat the system as having two degrees of freedom and use a Lagrange multiplier λ to enforce the constraint. Use plane polar coordinates as the two generalized coordinates. The polar angle θ will be measured from a y axis that points upwards so that θ is positive when the mass point is displaced away from its initial position. The other generalized coordinate will be the radial displacement r of the mass from the center of the hoop. The kinetic energy of the mass point is T = m 2 ( · x 2 + · y 2 ) . (1) In the polar coordinate system we have x = r sin θ , (2) y = r cos θ , (3) from which it follows that · x = · r sin θ + r · θ cos θ , (4) · y = · r cos θ r · θ sin θ , (5) and Eq.(1) can then be written as T = 1 2 m [ · r 2 + r 2 · θ 2 ] . (6) The potential energy may be taken as V = mgy = mgr cos θ . (7) With Eqs.(6) and (7), the Lagrangian for this system is

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