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$($a$,40pt)$ A thin walled tube is subjected to equal magnitude torques $T$ at both ends. The tube has an outer radius of $r,$ wall thickness $t,$ and shear modulus $G.$ Determine $($i$)$ the maximum and minimum shear stresses in the tube; $($ii$)$ maximum and minimal shear strains; $($iii$)($relative$)$ angle of twist; $($iv$)$ rate of twist. In total $6$ quantities.

$($b$,20$ bonus pt$)$ The same tube is slit lengthwise as shown in picture. Under the same torques $T,$ determine its $($i$)$ maximum shear stress and $($ii$)$ angle of twist. How do they compare to the original tube?

Hint: this is not a circular cross$-$section now due to the slit. Refer to textbook.

Part $1.$

i$,$ ii$,$iii and iv$.$

Part two $($b$)$

i and ii

In that order