HW6_prob1_probGPS33_G1

HW6_prob1_probGPS33_G1 - GPS 3.33 We can use a cylindrical...

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GPS 3.33 We can use a cylindrical coordinate system ( ρ , θ , z ) to describe the position of the particle. The polar ( z ) axis points upwards, ρ is the radial distance measured from the z axis ( ρ = p x 2 + y 2 ), and θ is measured in the x - y plane from the x axis. Because the particle is constrained to move on the surface of a paraboloid of revolution, this system has only two degrees of freedom. We will use ρ and θ as generalized coordinates to set up the Lagrangian after eliminating z and · z with the holonomic equation of constraint. In cylindrical coordinates the kinetic energy of a single particle is T = m 2 ( · ρ 2 + ρ 2 · θ 2 + · z 2 ) . (1) The potential energy for this system is V = mgz , (2) and the constraint equation is z = 2 , (3) where a is a constant that determines the shape of the paraboloid of revolution. From Eq.(3), it follows that · z =2 · ρ. (4) From these four equations we may write the Lagrangian in terms of ρ , · ρ ,and · θ as
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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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HW6_prob1_probGPS33_G1 - GPS 3.33 We can use a cylindrical...

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