Rest length of springs non$-$zero

If the rest length of the springs is not zero $($i$.$e$.L0),$ then the system of equations $(4)$ is

non$-$homogeneous. In this case, the solution will be given by the sum of the solution to the

homogeneous system and a particular solution:

$Z=Z_h+Z_p$

where $Z_h$ is the solution from Question $4$ and $Z_p$ is given by $Z_p=A^{-1}L.$ For Question $9$ and

Tasks $5-9,$ assume the springs both have rest length $l=2m.$

Task $5$ In your MatLaB script file, add the extra term $(A^{-1}L)$ to your solution for $Z.$

Task $6$ Create a MatLaB figure that plots $z_2(t)($y$-$axis$)$ against $z_1(t)(x-$axis$)$ for $0t40$

$($ie plot the position of the second mass against the position of the first$).$ Be sure to label your

axes and include a printout of this figure in your report $($Figure $3).$ This is an example of a

Lissajous curve. $($You do not need to know this, but you might be interested in the graph that

it creates!$)$

It is also interesting to calculate the velocity of the masses.

Question $9$ By differentiating your general solution, write down an expression for the velocity

of the masses $($write your answer in vector form$).$

Task $7$ Fill in the necessary MATLAB code required to calculate the velocities of the masses at

times $0t40.$

Task $8$ Create a MatLaB figure that plots the velocity of the first mass, $u_1(t)=d\frac{z_1}{d}t($y$-$axis$)$

against its position, $z_1(t)($x$-$axis$)($Figure $4).$

Task $9$ Create a MatLaB figure that plots the velocity of the second mass, $u_2(t)=d\frac{z_2}{d}t$

$($y$-$axis$)$ against the position of the first, $z_1(t)($x$-$axis$)($Figure $5).$