Assignment:
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.
(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.
Provide complete and step by step solution for the question and show calculations and use formulas.