Physics Friday 27

Last week, we discussed the one-dimensional problem of two identical masses connected to each other by a spring of constant k2, and to fixed walls by springs of constant k1, with the springs all at resting length at equilibrium. Note that the resting length of the springs was not a factor in the computations or the solution. Suppose we extend the line to three masses instead of two, giving us two springs of constant k2.

What is the behavior of these masses for small displacements along the line?

In analogy to the previous problem, let the position of the masses be given by x1, x2, and x2, with each being zero at the initial positions, and positive values for rightward displacement. Using ΣF=ma, we have equations of motion:
,
,
and
.

We again let  and , and use matrices to combine the equations. Doing this, we have:
.

Here, we again attempt a solution of the form
.
Substituting, we obtain:


So again we have that  is an eigenvector of the matrix with eigenvalue r2.
Using: , we find the equation for the eigenvalues x to be:

This gives us eigenvalues:
,
,
,,
all of which are negative (as ). Thus we have oscillatory behavior, with angular frequencies , , and  (in increasing order).

Finding the corresponding eigenvectors, we see they are , , and .

We note that for v1, all three terms are positive; for v2, the middle mass is stationary and the other two move in opposite directions; and for v3, the middle mass moves opposite the other two.

We can express this as changes of phase as we move from left to right: the lowest mode has zero changes in phase; the next mode has one change in phase, at the middle mass; the highest mode has two changes of phase, between each mass. Compare this to last week’s result, when we had two modes: the lower-frequency mode had no phase changes, the higher one had one phase change.