The energy of the mass$-$spring system is given by the sum of the kinetic energy and the

potential energy. In the absence of damping, the energy is conserved.

$($a$)$ Add commands to LAB$05$ex$1$ to compute and plot the quantity $E=\frac{1}{2}m{v}^2+\frac{1}{2}k{y}^2$

as a function of time. What do you observe? $($pay close attention to the $y-$axis scale

and, if necessary, use ylim to get a better graph$).$ Include at least one plot. Does

the graph confirm the fact that the energy is conserved?

$($b$)$ Show analytically that $\frac{dE}{dt}=0.($Note that this proves that the energy is constant$).$

$($c$)$ Add commands to LAB$05$ex$1$ to plot $v$ vs $y($phase plot$).$ Include the plot. Does the

curve ever get close to the origin? Why or why not? What does that mean for the

mass$-$spring system?LAB$05$ex$1.$m

clear all; $\%$ clear all variables

$m=16$;$\%$mass$[kg]$

$k=49$;$\%$ spring constant $\frac{N}{m}$

omega$0=\sqrt[\phantom{2}]{\frac{k}{m}}$;

y$0=-0\mathrm{.}8$; v$0=,0\mathrm{.}3$;$\%$ initial conditions

$,$ omega$0)$; solve for $y=Y($:$,1)$;$v=Y($:$,2)$;$\%y,VY\%y=Y(1)$;$V=Y(2)=[V$;$\}\{$:${?}^{{?}^{{?}^{?}}}2**y]0$

$y=Y($:$,1)$;$v=Y($:$,2)$;$\%$ retrieve $y,V$ from $Y$

figure$(1)$; plot$(t,y,\text{'}ro-\text{'},t,v,\text{'}b+-\text{'})$;$\%$ time series for $y$ and $v$

grid $on$; axis tight;

$\%---------------------------------------------------------$

function $dYdt=f(t,Y,$ omegaO$)$; $\%$ function defining the $DE$

$y=Y(1)$;$V=Y(2)$;

$dYdt=[V$;$\}$ omegaO$\{$:${?}^{{?}^{{?}^{?}}}2**y]$;

end