Question #$5$: Two$-$fluid U$-$tube Manometer.

Using pressure balances on the fluids found in the figure below, derive the given equation

explicitly.

$p_a-p_b=(R-R_0)({}_A-{}_B+\frac{a}{A}{}_B-\frac{a}{A}{}_A)g$

Let $A{m}^2$ be the cross$-$sectional area of each of the large

reservoirs and $m^2$ be the cross$-$sectional area of each of the

tubes forming the $U.$

$R_0$ is the manometer reading when $p_a=p_b$

$R$ is the actual manometer reading

${}_A$ is the density of the heavier fluid

${}_{}$ is the density of the lighter fluid