Question $4(30$ points$)[$Application of energy equation$]$

Oil flows in an inclined pipe at a rate $Q$ as shown in the figure. The differential reading in the mercury manometer is $L.$ The specific gravity of oil and mercury are $S{G}_{\text{o}\text{i}\text{l}}$ and $S{G}_{Hg},$ respectively.

a$)$ Derive the equation to compute the shaft work head, $h_s,$ that the pump supplies to the oil if head losses are negligible. $(25$ points$)$

Note: simplify your equation as much as possible.

b$)$ Determine the magnitude of the shaft work head, $h_s,$ if the following magnitudes are known

$S{G}_{\text{o}\text{i}\text{l}}=0\mathrm{.}88$

$S{G}_{Hg}=13\mathrm{.}6$

$A_l=0\mathrm{.}785f{t}^2$

$A_2=0\mathrm{.}196f{t}^2$

$Q=5f\frac{t^3}{s}$

$L=3ft$

$h=2ft$

$H=2ft$