Question

Two immiscible liquids of densities $\rho$ and $2\rho$ and thickness (heights) $h$ and $2h$, respectively, push a vertical plate. Find the total side thrust given by the liquids on the vertical plate.

Answer 1

Let small force exerted be

$dF=p.b.dh$

Pressure$=P=xpg$

$x=$depth

Then total force exerted$=\int dF$

$={\int }_0^{h}xpgb.dx+{\int }_0^{2h}(hpg+x.2pg)b.dx$

$=b.p.g{\left[\frac{x^2}{2}\right]}_0^h+{\left[hpgbx\right]}_0^{2h}+2pgb{\left[\frac{x^2}{2}\right]}_0^{2h}$

$=b.p.g[\frac{h^2}{2}+2h^2+4h^2]$

$=b.p.g\left[\frac{h^2+4h^2+8h^2}{2}\right]$

$F=\frac{13}{2}bpg{h}^2$