Question $5$: An element in plane stress is subjected to the normal and shear stresses as shown. Use Mohr's circle to do the following:

$($i$)$ determine the principal stresses and orientation of the principal planes, showing your answer on a sketch of a properly oriented element.

$($ii$)$ determine the maximum in$-$plane shear stress $($and associated normal stresses$)$ and orientation of the corresponding planes. Once again, show your answer on a sketch of a properly oriented element.

$($iii$)$ what is the reason you need to calculate principal stresses and$/$or the maximum shear stress $($not more than $10$ words of explanation please$)$

Note that credit will only be given for a solution using Mohr's circle. Stress transformation equations can only be used to check your answer.