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HW13_prob3_GPS20_G10

HW13_prob3_GPS20_G10 - GPS Problem 8.20(2nd ed 8.10 Modied...

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GPS Problem 8.20 (2 nd ed. 8.10) Modi fi ed Planar Double Pendulum This problem is a variation of the planar double pendulum problem, illustra- trated in Fig. 1.4. The solution to that problem, GPS 1.22 or 2 nd ed. 1.20, will be useful here. There are two major modi fi cations that are required. First, we put m 1 = 0 and m 2 = m . Second, the angle θ 1 is now a driven coordinate with a uniform rotation frequency, that will be denoted as ω . So we have · θ 1 = ω and θ 1 ( t ) = ωt , with θ 1 (0) = 0 for simplicity. The remaining changes are minor: we let l 1 = a , l 2 = l , and θ 2 = θ . Because the motion of the mass point is con fi ned to the vertical plane, the system has only one degree of freedom, described by the angle θ , θ 2 in Fig. 1.4, as the generalized coordinate. The angles will be measured from a y axis that points downwards so that θ 1 and θ 2 are positive as shown in Fig. 1.4. The kinetic energy of the pendulum is T = m 2 ( · x 2 + · y 2 ) . (1) In the polar coordinate system we have x = a sin ωt l sin θ, (2) y = a cos ωt + l cos θ, (3)

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