Why Do Different Normal Frequencies Yield Multiple Amplitude Configurations?

[rmfw]

Homework Statement

(see pic)

$\frac{l_{2}}{l_{1}} = \frac{1}{4}$
$\frac{g}{l_{1}} = 1$

I need to find the normal modes of oscillation. (for small oscillations)

The Attempt at a Solution

I solved the problem using the matricial way and got the following matrix: (I simplified it using the above ratios)$V - wT = m \begin{pmatrix}<br /> 1 - w^2 & -1/4 \\<br /> -1/4 & 1/8 - w^2/48<br /> \end{pmatrix} <br />$Solving the determinant of the matrix I got two positive solutions. Now the thing is for each solution I got two different configurations of amplitudes. I thought that each normal frequency would give me only one configuration of amplitudes. Can you make this clear for me? Thanks.

 

[BruceW]

Hi, sorry but what was the problem statement? where is the mass located? It is hard to follow along without knowing that...