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Past exam of the mathematical methods II course of the University of Oxford

2017

The squared-frequency matrix is
$$
\frac gl I+\frac km\begin{pmatrix}1&-1&0\\-1&2&-1\\0&-1&1\end{pmatrix}.
$$
The dimensionless spring matrix has an <orthonormal eigenbasis>
$$
v_1=\frac1{\sqrt3}(1,1,1),\quad
v_2=\frac1{\sqrt2}(1,0,-1),\quad
v_3=\frac1{\sqrt6}(1,-2,1),
$$
with <eigenvalues> $0,1,3$. Adding the gravitational term therefore gives
$$
\boxed{\omega_1=\sqrt{\frac gl},\quad
\omega_2=\sqrt{\frac gl+\frac km},\quad
\omega_3=\sqrt{\frac gl+\frac{3k}m}.}
$$
The uniform mode leaves both springs unstretched. The other two modes deform the springs and so have higher frequencies.

Solution

Past exam of the mathematical methods II course of the University of Oxford / 2017 / 9 / c / Solution

Past exam of the mathematical methods II course of the University of Oxford / 2017 / 9 / c / Solution

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Past exam of the mathematical methods II course of the University of Oxford / 2017 / 9 / c / Solution

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