Problem $5.$ A staft is simply supported at A and $B$ as shown. A constant magnitude transverse load $P$ is applied as the shaft rotates subjected to a time$-$varyiag torque from $T_{\text{m}\text{i}\text{s}}$ to $T_{\text{m}\text{e}\text{x}\text{.}}.$ If $\frac{?}{?}=12in,a=7in,b=9in,P=500lb,T_{\text{m}\text{i}\text{n}}=-100lb-in,T_{\text{m}\text{a}\text{x}}=600lb-$in$.$ Find the shaft diameter requirod to obtain a safety fnetor of $2$ in fatigua lasding, and the corresponding maximum deflection of the shaft. Assume a machined surface, reliability of $99\%$ and room temperature, use $S_{el}=108$ksi,$S_j=62$ksi, and $E=2\mathrm{.}9{10}^4$ksi.

$(20$ marks$)$

assume bearings act as simple supperts