$4($a$)$ A solid circular disc of outside radius $b=400mm$ is bored to allow a shrink fit $($or interface fit$)$ onto a solid shaft. The shaft$/$disc interface after assembly is at a radius $250$ mm from the centre of the shaft. If $=0\mathrm{.}1mm$ is the initial interference $($or overlap$),$ determine the interface pressure between the shaft and the disc, assuming the shaft is made of steel and the disc is made of aluminium. The properties for steel are $E=208$ GPa, $v=0\mathrm{.}33,$ and the density $=8000k\frac{g}{m^3},$ and those for aluminium are $E=70$ GPa, $v=0\mathrm{.}34,$ and density $=2700k\frac{g}{m^3}.$

$($b$)$ Determine the maximum torque the disc in part $($a$)$ can transmit if it has a thickness of $40$ mm parallel to the longitudinal axis of the shaft, and the coefficient of friction between the disc and the shaft is $0\mathrm{.}3.$