A bead of mass m moves on a parabolic wire with equation z=(1/2*x^2), where z measures the height of the bead and x is the horizontal distance in the plane of the wire. The plane in which the wire lies rotates about a fixed vertical axis passing through x = 0, so that its angle relative to a fixed plane is ø. (See attachment for diagram).
(a) Show that the kinetic and potential energies of the bead are respectively, T = m/2 [(1+x^2)x^2+x^2ø^2] and V=(m/2)*gx^2.
(b) Write down the Lagrangian for this system, and hence derive the equations of motion. Show that the equation of motion for ø implies that x^2ø=K, where K is a constant. Hence obtain an equation of motion for x that does not contain ø or its derivatives.
(c) Show that there is a solution of the equations of motion where x takes a constant value, x_0. Show that: x_0=(K^2/9)^1/4.
(d) There also exists a solution where x(t) makes small oscillations. By substitution x(t)= x_0 + Esin(wt) into the equation of motion for x and neglecting terms of order E^2 and higher, determine the angular frequency of these small oscillations in terms of K and g.
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The solution is attached below in two files. the files are identical in content, only differ in format. The ...
The detailed solution shows how to construct the Lagrangian, extract the equations of motion and use perturbation theory to find the small oscillations solution.