Two Magnets and a Spring Revisited In this lab we again study the motion of a mass that...

Two Magnets and a Spring Revisited

In this lab we again study the motion of a mass that can slide on the x-axis (see Lab 2.1). The mass is attached to a spring that has its other end attached to the point (0, 2) on the y-axis. In addition, the mass is made of iron and is attracted to two magnets of equal strength—one located at the point (−1,−a) and the other at (1,−a) (see Figure 4.40).

We assume that the spring obeys Hooke’s Law, and the magnets attract the mass with a force proportional to the inverse of the square of the distance of the mass to the magnet (the inverse square law). In Lab 2.1, we observed a subtle dependence of the solutions on the position of the magnets.

In this lab, we consider the effect of an external forcing term on this system. We can think of the external force as a wind that gusts, alternately blowing the mass to the left and the right. More precisely, we study the solutions of the nonautonomous second-order equation

where b is the amplitude of the forcing. This equation is very complicated, so you are expected to carry out your analysis numerically. In order to observe the effects of the forcing, you will have to follow the solutions over intervals of time that are at least as long as several periods of the forcing function. Address the following items in your report:

In your report, pay particular attention to the physical interpretation of the solutions in terms of the possible motions of the mass as it slides along the x-axis. Include graphs and phase portraits to illustrate your discussion, but pictures alone are not sufficient.

Study solutions of the forced system for magnets that are close to the x-axis assuming the amplitude of the forcing is small, for example, a = 0.5 and b = 0.1. How do solutions differ from the unforced case? Express your conclusions in terms of the solution curves in the phase plane and in terms of the motion of the mass along the x-axis. Pay particular attention to solutions whose initial conditions are near the origin in the phase plane.