HW3_prob4_Ch2GPS18_G17

HW3_prob4_Ch2GPS18_G - Goldstein Problem 2.17(3rd ed 2.18 The geometry of the problem A particle of mass m is constrained to move on a circular

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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 f erence is that here we will let the polar ( z ) axis point upwards (see the f gure) so that (1) the polar angle θ di f ers by π from the earlier de f 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 f 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 .A sare su l t , ϕ = ω t
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This note was uploaded on 12/19/2010 for the course PHYSICS ph 409 taught by Professor Wilemski during the Fall '10 term at Missouri S&T.

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HW3_prob4_Ch2GPS18_G - Goldstein Problem 2.17(3rd ed 2.18 The geometry of the problem A particle of mass m is constrained to move on a circular

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