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Equation of motion of a bead on a rotating parabola

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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 detailed solution shows how to construct the Lagrangian, extract the equations of motion and use perturbation theory to find the small oscillations solution.

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Consider a bead of mass m sliding without friction on a wire that is bent in the shape of a parabola and is being spun with constant angular velocity w about its vertical axis, as shown in figure 7.17. Use cylindrical polar coordinates and let the equation of the parabola be z=kp^2. Write down the Lagrangian in terms of p as the generalized coordinate. Find the equation of motion of the bead and determine whether there are positions of equilibrium, that is, values of p at which the bead can remain fixed, without sliding up or down the spinning wire. Discuss the stability of any equilibrium position you find.

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