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Statement of a problem № 3451

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A particle of mass m is constrained to move on an ellipse E in a vertical plane, parametrized by x = a cos T , y = b sin T , where a, b > 0 and a = b and the positive y-direction is the upward vertical. The particle is connected to the origin by a spring, as shown in the diagram, and is subject to gravity. The potential energy in the spring is 1 kr 2 where r is the distance of the point mass from the origin (Fig. 7.10). (i) Using T as a coordinate, find the kinetic and potential energies of the particle when moving on the ellipse. Write down the Lagrangian and show that Lagrange's equation becomes m(a^2 sin^2 T + b^2 cos^2 T)(T··) = (a^2-b^2 )(k-m(T··)^2 ) sin T cos T-mgb cos T . (ii) Show that T = A±pi/2 are two equilibrium points and find any other equi- librium points, giving carefully the conditions under which they exist. You may either use Lagrange's equation or proceed directly

Solution:
A particle of mass m is constrained to move on an ellipse E in a vertical plane,

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