A Mass-Spring System with a Rubber Band In an idealized mass-spring system, the spring...

A Mass-Spring System with a Rubber Band

In an idealized mass-spring system, the spring provides a restoring force proportional to its displacement from its rest position (Hooke’s Law). This ideal spring is never overstretched or completely compressed. These sorts of assumptions (hopefully) yield models that are simple enough to be tractable but accurate enough to be useful. But we should always be mindful of the underlying assumptions.

For example, we frequently think of a rubber band as a kind of spring. When stretched, it provides a restoring force toward its rest length. However, there are limitations. We cannot stretch the rubber band too far or it will break. More importantly, a compressed rubber band provides no force. We expect the behavior of a system with a spring and a rubber band to be different from one with a spring alone.

In this lab we compare models for a mass-spring system and a mass-spring system with the addition of a rubber band (see Figure 2.68). The rubber band adds extra restoring force when the displacement is positive but adds no force when the displacement is negative.

The mass-spring system depicted in Figure 2.68 is modeled by

where y measures the vertical displacement (with down as positive) in meters. The parameters are the mass m, the damping coefficient b, and the spring constant k₁. The constant 10m on the right-hand side of the equation is a rough approximation of the force due to gravity.

To include the rubber band, we add an extra term to the equation above. We assume that the rubber band obeys Hooke’s Law when it is stretched but that it exerts no

force when it is compressed. Let h(y) be the function that is y if y is positive and zero if y is negative, that is,

Your report: Address each of the previous items. You may provide illustrations from the computer, but remember that although a good illustration is worth 1000 words, 1000 illustrations are worth nothing. Make sure you use your conclusions about the solutions of these equations to describe how the mass oscillates.

(Ideal mass-spring system with no rubber band) Choose a value of k₁ such that

12 < k₁ < 13 and study solutions of the equation

Examine solutions using both their graphs and the phase portrait. Are solutions periodic? If so, approximate the period of the solutions. Be specific about the physical interpretation of the solutions for different initial conditions.