A bead with mass m slides frictionless on a massles parabolic wire. Its position

Question

A bead with mass m slides frictionless on a massles parabolic wire. Its position on the wire is measured by x in a frame of reference fixed to the wire as shown in the figure below. The wire is driven to rotate periodically about its axis of symmetry, which is parallel to the direction of gravity as shown below. The angular velocity of the wire is given by A sin(omega t). Find the ordinary differential equation in x(t) which describes the motion of the bed relative to the wire-fixed reference frame.

Explanation / Answer

Finding the lagrangian L=0.5*m((dx/dt)^2 + (x^2)*(w^2) + 4(c^2)*(x^2)*(dx/dt)^2))-mgc(x^2) px = dL/(dx*) = m*(dx/dt) + 4*(c^2)*m*(x^2)(dx/dt)) (dpx)/(dt) == dL/dx = m*x*(w^2) + 4*(c^2)*m*(x)(dx/dt)^2)-2mgc(x)

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