= Solution
For $\lambda=1$, the first <eigenvector> equation gives $y_1=y_2$. For $\lambda=4$, it gives $y_1=-2y_2$. Therefore
$$
\boxed{\mathbf a_-\propto(1,1),\qquad \mathbf a_+\propto(2,-1).}
$$
The lower mode moves the masses in phase with equal displacements. In the higher mode they move in opposite directions, with the lighter mass's amplitude twice that of the heavier mass. Their <eigenvectors> satisfy <mass-weighted orthogonality> with respect to the physical <mass matrix> $\operatorname{diag}(m,2m)$, although their ordinary Euclidean <dot product> is nonzero.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2016-resit-two-mass-normal-modes.png]
{title=Normal-mode shapes and an initially displaced light mass on a string with two point masses}
Back to article page
