Question 8 (Unit 11) - 6 marks
The oscillations of three particles A, B and C are governed by the matrix equation
$\begin{pmatrix} \ddot{x} \\ \ddot{y} \\ \ddot{z} \end{pmatrix} = \begin{pmatrix} -3 & 1 & 0 \\ 2 & -8 & 6 \\ 0 & 3 & -3 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}$,
where $x$, $y$ and $z$ are the particles' respective displacements from equilibrium.
(a) Show that the vector $(1 \ 2.62 \ 3)^T$ is an approximate eigenvector of the dynamic matrix (to two significant figures accuracy) by calculating the corresponding eigenvalue.
(b) Calculate the corresponding normal mode angular frequency for the normal mode with the given normal mode eigenvector.
(c) Give an initial displacement vector that would lead to the normal mode motion corresponding to the given eigenvector.
(d) For this normal mode motion, state (with a reason) which of the particles are moving in phase.