Question $1$

Write out the $($relevant$)$ equations of motion for the following system $($shown in the figure

belowy$)$ when:

$($a$)$ A and B are boch free. $($b$)$ A is clamped. $($c$)$ B is clamped. $($Assume A moves along the line

of the springs.$)$ Note: You do NOT have to solve these equations of motion.

Question $2$

Two objects, A and B$,$ each of mass m$,$ are connected by springs as shown in the figure below.

The coupling spring bas a spring constant $k,$ and the other two springs hawe spring coestant $k.$

If is clamped, $$ vibrates at a frequency $v_A$ of $1\mathrm{.}81se{c}^{-1}.$ The frequency $v_1$ of the lower normal

mode is $1\mathrm{.}14se{c}^{-1}.$

$($a$)$ Write down the equations of motion of A and $B.($b$)$ Putting ${}_0=\sqrt[\phantom{2}]{\frac{k_0}{m}},$ show that the

angular frequencies ${}_1$ and ${}_2$ of the normal modes are given by

${}_1={}_0$ and $,{}_2=\sqrt[\phantom{2}]{{}_0^2+2\frac{k_c}{m}}$

and that the angular frequency of A and $B$ is clamped $(x_B=0$ always$)$ is given by

${}_A=\sqrt[\phantom{2}]{{}_0^2+\frac{k_c}{m}}$

$($c$)$ Using the numerical data above, calculate the expected frequency $(v_2)$ of the higher normal

mode. $($The observed value was $2\mathrm{.}27se{c}^{-1}).$ From the same data calculate the ratio $\frac{k_c}{k_0}$ of

the two spring constants.

Question $3$

$($a$)$ Two objects of equal mass $m$ are attached to two opposing walls by two identical springs

of spring constant $2k$ and coupled by a third of spring constant $k($as illustrated in the figure

Assignement questions and problems $-$

PHY$2606\_0\_2023$

below$).$ By explicit consideration of the forces on each object show that the equations of

motion of the two obiects are: $m{x}^{}=-3kx+kv$ and $m{v}^{}=-3kv+kx.$

$($b$)($i$)$ Explain how to excie the system to vibrate in each of the two normal modes. $($ii$)$

Describe the motion of the masses in the two normal modes. $($iii$)$ Explain why the low$-$

frequency mode is independent of the coupling spring.

Ouestion $4$