A $60$ strain rosette is mounted on the surface of the compressor pump, at point $A,$ to monitor the loading and deformation of the component. The following readings are obtained from each gauge:

${}_a=-300({10}^{-6}),{}_b=-500({10}^{-6}),$ and ${}_c=$ last three numerical digits of your student PID $({10}^{-6})$

Consider ${}_c$ equal to your student PID's last three numerical digits multiplied by ${10}^{-6}.$ For example, if your PID is A$18752350,$ then ${}_c=350({10}^{-6})$

Within the plane that the strain rosette is installed, determine

$($a$)$ state of strain,

$($b$)$ the state of stress $($all stresses$),$

$($c$)$ the principal stresses

Assume $E=160$ GPa and $v=0\mathrm{.}3$

$(15$ points$)$

Note: If needed, here are the general equations for strain rosettes $($you may use your customized equation for this rosette$).$

${}_a={}_{x}co{s}^2{}_a+{}_{y}si{n}^2{}_a+{}_{xy}sin{}_{a}cos{}_a$

${}_b={}_{x}co{s}^2{}_b+{}_{y}si{n}^2{}_b+{}_{xy}sin{}_{b}cos{}_b$

${}_c={}_{x}co{s}^2{}_c+{}_{y}si{n}^2{}_c+{}_{xy}sin{}_{c}cos{}_c$