$(20$ pts$)$ Consider the cantilever beam subject to a downward load force at its outer end as shown in the figure below. Two strain gauges $($ a and $b)$ are attached to the top of the beam where they are stretched, increasing their resistance by DR when the load is applied. The other two strain gauges $($c and d$)$ are attached to the bottom of the beam where they are compressed, decreasing their resistance by DR$.$ The change in resistance is governed by: $R=R0G,$ where $G$ is the gauge factor $($which is typically a constant number close to $2\mathrm{.}0),R0$ is the resistance of the strain gauge at its relaxed state, and $$ is the mechanical strain. The gauge positions $c$ and $d$ are opposite of positions a and $b$ but at the bottom side of the cantilever.

For the four strain gauges $($a$,$ b$,$ c$,$ and d$),$ you have four potential places in your deflectiontype DC bridge $(1,2,3,$ and $4).$ Find the bridge configuration rules $($i$.$e$.,$ map a$-$d to $1-4,$ which resistors from $1\mathrm{.}4$ should be on the same side of the beam, which resistors should be on the opposite sides of the beam$)$ that give you a linear relationship between strain and output voltage. Also, find that linear relation.