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HW3_prob4_Ch2GPS18_G17

# HW3_prob4_Ch2GPS18_G17 - Goldstein Problem 2.17(3rd ed 2.18...

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C Goldstein Problem 2.17 (3 rd ed. # 2.18) The geometry of the problem: A particle of mass m is constrained to move on a circular hoop of radius a that is vertically oriented and forced to rotate about the vertical symmetry axis with angular frequency ω . The only external force is that of gravity. The symmetry of this problem strongly suggests the use of spherical coordinates ( r , θ , φ ), defined in the figure, as the generalized coordinates: r is the radial displacement of the particle from the origin, θ is the polar angle, and φ is the azimuthal angle. You could also use cylindrical coordinates ( ρ , φ , z ), where ρ 2 = x 2 + y 2 , but the analysis is then a little more complicated.

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Goldstein 2-17 (3 rd ed. #2.18) The Lagrangian of this system is essentially that of the spherical pendulum, which we worked out in Problem GPS 1.19. One di ff erence is that here we will let the polar ( z ) axis point upwards (see the fi gure) so that (1) the polar angle θ di ff ers by π from the earlier de fi nition and (2) the potential energy V will be opposite in sign to that of the earlier problem. The radial coordinate r is, of course, constant and equals the radius of the hoop a . Hence, r is not a generalized coordinate, i.e., there is no radial degree of freedom. The other principal di ff erence between this problem and the spherical pendulum is that the azimuthal angle ϕ is not a generalized coordinate in this problem because · ϕ is prescribed as · ϕ = ω . As a result, ϕ = ω t + ϕ 0 , and we lose another degree of freedom. Thus, we see that this system has only one degree of freedom that
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