Two Magnets and a Spring In this lab we consider the motion of a mass that can slide fr...

Two Magnets and a Spring

In this lab we consider the motion of a mass that can slide freely along the x-axis. 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 2.67).

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). If we choose the spring constant, mass, strength of the magnets, and units of distance and time appropriately, then we can model the motion of the mass along the x-axis with the equation

(A good exercise for engineering and physics students: Derive this equation and determine the units and choices of spring constant, rest length of the spring, mass, and strength of the magnets involved.)

The goal of this lab is to study this system numerically. Use technology to find equilibria and study the behavior of solutions. Be careful to consider the correct regions of the phase plane at the correct scale so that you can find the important aspects of the system.

In your report, you should address the following items:

Your report: Address each of the items above. 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. You may include graphs and phase portraits to illustrate your discussion, but pictures alone are not sufficient.

Consider the system with the parameter value a = 0.5. Discuss the behavior of solutions in the phase plane. Relate the phase portrait to the possible motions of the mass along the x-axis. Be particularly careful to describe the solutions that separate different types of qualitative behavior.