Question $2$: Four gears are attached to a circular shaft and transmit the torques shown in the figure. The allowable shear stress in the shaft is $10,000.$

a$)$ Calculate the required diameter of the shaft $d$ if it has a solid cross section. Plot the torsional moment diagram $($TMD$)$ and the torsional displacement diagram $($TDD$)$ for the entire shaft.

b$)$ Calculate the required outside diameter $d$ if the shaft is hollow with an inside diameter of $1\mathrm{.}0$ in $.$ Plot the torsional moment diagram $($TMD$)$ and the torsional displacement diagram $($TDD$)$ for the entire shaft.