A Current Interest. The motion of an iron bar attracted by

QUESTION:

A Problem of Current Interest. The motion of an iron bar attracted by the magnetic field produced by a parallel current wire and restrained by springs (see Figure  ) is governed by the equation

                                          \(\frac{d^{2} x}{d t^{2}}=-x+\frac{1}{\lambda-x}\) for

                                           \(-x_{0}<x<\lambda\)

       Figure 5.17 Bar restrained by springs

         and attracted by a parallel current

where the constants \(x_{0}\) and \(\lambda\) are, respectively, the distances from the bar to the wall and to the wire when the bar is at equilibrium (rest) with the current off.
(a) Setting \(v=d x / d t\), convert the second-order equation to an equivalent first-order system.
(b) Solve the phase plane equation for the system in part (a) and thereby show that its solutions are given by

                                             \(v=\pm \sqrt{C-x^{2}-2 \ln (\lambda-x)}\)

where \(C\) is a constant.
(c) Show that if \(\lambda<2\) there are no critical points in the  -phase plane, whereas if \(\lambda>2\) there are two critical points. For the latter case, determine these critical points.
(d) Physically, the case \(\lambda<2\) corresponds to a current so high that the magnetic attraction completely overpowers the spring. To gain insight into this, use software to plot the phase plane diagrams for the system when \(\lambda=1\) and when \(\lambda=3\).
(e) From your phase plane diagrams in part (d), describe the possible motions of the bar when \(\lambda=1\) and when \(\lambda=3\), under various initial conditions.

Equation Transcription:

Text Transcription:

d^2x/dt^2=-x+1/lambda-x,

-x_0<x<lambda

x_0

lambda

v=dx/dt

v=pm sqrt C-x^2-2ln⁡(lamda-x)

C

lambda <2

lambda >2

lambda =1

lambda =3