$77.**$ Two blocks of masses $\backslash ($ m$\_\{1\}\backslash )$ and $\backslash ($ m$\_\{2\}\backslash )$ are connected with a string that passes over a very light pulley $($see Figure $\backslash (\backslash$mathrm$\{$P$\}\_\{7\mathrm{.}77\}\backslash ).).$ Friction in the pulley can be ignored. Block r is resting on a rough table and block $2$ is hanging over the edge. The coefficient of friction between the block I and the table is $\backslash (\backslash$mu $\backslash )($assume static and kinetic friction have the same value$).$ Block I is also connected to a spring with a constant $\backslash ($ k $\backslash ).$ In the initial state, the spring is relaxed as a person is holding block $2,$ but the string is still taut. When block $2$ is released, it moves down for a distance $\backslash ($ d $\backslash )$ until it stops $($the final state$).($a$)$ Explain why block $2$ eventually stops moving. $($b$)$ Choose a system and represent the process with a work$-$energy bar chart. $($c$)$ Derive the expression for distance $\backslash ($ d $\backslash )$ in terms of $\backslash ($ m$\_\{1\},$ m$\_\{2\},\backslash$mu $\backslash ),$ and $\backslash ($ k $\backslash ),$ and evaluate the expression using extreme case analysis and unit analysis.

Figure $\backslash (\backslash$mathrm$\{$P$\}\_\{7\mathrm{.}77\}\backslash )$

AMMOH $1$