$1\mathrm{.}2$ Problems Part $2$

$4$ masses are connected in series to $4$ springs with K$=100$N$/$m$.$ What are the final positions of the masses?

Springs$-$masses

The masses haves the following amounts, $1,2,3,$ and $4$ kg for masses $1-4.$ Using a FBD for each mass:

$m$\_\{1\}$g$+$k$($x$\_\{2\}-$x$\_\{1\})-$kx$\_\{1\}=0$$

$m$\_\{2\}$g$+$k$($x$\_\{3\}-$x$\_\{2\})-$k$($x$\_\{2\}-$x$\_\{1\})=0$$

$m$\_\{3\}$g$+$k$($x$\_\{4\}-$x$\_\{3\})-$k$($x$\_\{3\}-$x$\_\{2\})=0$$

$m$\_\{4\}$g$-$k$($x$\_\{4\}-$x$\_\{3\})=0$$

in matrix form K$=100$ N$/$m:

$$\backslash$left$[\backslash$begin$\{$array$\}\{$cccc$\}2$k & $-$k & $0$ & $0\backslash \backslash -$k & $2$k & $-$k & $0\backslash \backslash 0$ & $-$k & $2$k & $-$k $\backslash \backslash 0$ & $0$ & $-$k & k $\backslash$end$\{$array$\}\backslash$right$]\backslash$left$[\backslash$begin$\{$array$\}\{$c$\}$ x$\_\{1\}\backslash \backslash$ x$\_\{2\}\backslash \backslash$ x$\_\{3\}\backslash \backslash$ x$\_\{4\}\backslash$end$\{$array$\}\backslash$right$]=\backslash$left$[\backslash$begin$\{$array$\}\{$c$\}$ m$\_\{1\}$g $\backslash \backslash$ m$\_\{2\}$g $\backslash \backslash$ m$\_\{3\}$g $\backslash \backslash$ m$\_\{4\}$g $\backslash$end$\{$array$\}\backslash$right$]$$