Problem $6(4$ points$)$: Figure $2$ shows a bridge circuit with two strain gauges applied on opposite sides of a beam in bending by a moment $M=FL.$ Assume $R_1=R_2=R,$ and when the load is zero, $R_3=R_4=R.$

a$.$ Use the bridge equation to derive an expression that relates the change in the output voltage $e_{\text{o}\text{u}\text{t}}$ to the bending strain ${}_{\text{b}\text{e}\text{n}\text{d}}={}_4=-3.$

b$.$ Let $w$ and $t$ be the width and thickness of the beam, respectively. L is distance from the centers of the strain gages to the load F$,$ and E is Young's modulus. Derive an expression that relates $e_{\text{o}\text{u}\text{t}}$ to the load, F $,$ at the end of the beam.

c$.$ Develop a numerical expression for the slope $e_{\text{o}\text{u}\text{t}}$ per unit of force $F$ if $e_{in}=2V,GF=2\mathrm{.}1,w=20mm,t=2mm,$ and $L=150mm.$ Assume the material is steel with E as discussed in class. Express result in units of microvolt per Newton $(\frac{V}{N}.$

Figure $2.$ Two strain gages arranged in a half$-$bridge circuit $($left$).R_3$ and $R_4$ are two gages sensing equal and opposite strains on the beam $($right; gage size exaggerated$).$