$($a$)$ A $50$ mm diameter tube gradually expands to $100$ mm diameter tube in a length of $10$ m $.$ If the tube makes an angle of $20$ in upward direction with the horizontal $($Fig$.2),$ determine the pressure $p_2$ at section $2$ if the tube carries a discharge of $3\mathrm{.}925\frac{l}{s}$ and the inlet pressure $p_1$ is $60k\frac{N}{m^2},$ assuming $($i$)$ no energy loss, and $($ii$)$ a loss of $0\mathrm{.}20$ m $.$

$($b$)$ Derive Darcy$-$Weisbach formula for head loss in pipe due to friction.