b
Consider an element of the liquid of height \(dh\) and area of cross\(-\)section \(\mathrm{A}\) at a depth \(\mathrm{h}\) below the surface of the liquid. Let \(p\) and \(p\) \(+\) \(\mathrm{dp}\) be the liquid pressures at the upper and lower surfaces of the element. Mass of the liquid in the element, \(\mathrm{dm}=\) \(Adh\)\(\rho\)

Net upward force on the element

\(=[(p+d p) A-p A]-g d m\)

Or \(\mathrm{A} \mathrm{dp}-\mathrm{g} \mathrm{dm}=\mathrm{a} \mathrm{dm}\)

(As the element moves up with an acceleration a)

Or \(\mathrm{A} \mathrm{dp}=(\mathrm{g}+\mathrm{a}) \mathrm{dm}=(\mathrm{g}+\mathrm{a})\) \(Adh\rho.\)

\(\int \mathrm{d} \mathrm{p}=\int \rho(\mathrm{g}+\mathrm{a}) \mathrm{dh}\) Or \(\mathrm{p}=\rho(\mathrm{g}+\mathrm{a}) \mathrm{h}\)