QUESTION $4$

$10$ Marks

Derive an expression for the pressure variation in a liquid where the specific weight increases with depth, $h,$ according to the relation $=kh+{}_0,$ where k is a constant and ${}_0$ is the specific weight at the free surface.

QUESTION $5$

$10$ Marks

Small differences in gas pressure are often measured in micromanometer, as shown in figure $5$ below. This device comprises two large reservoirs, each with a cross$-$sectional area $A_t,$ filled with a liquid of specific weight ${}_1.$ These reservoirs are connected by a $U-$tube with a cross$-$sectional area $A_d,$ containing a liquid of specific weight, ${}_2.$ To ensure that this reading is sufficiently large for easy measurement of small pressure differentials, determine the relationship between $h$ and $p_1-p_2$ when the ratio $\frac{A_1}{A_1}$ is small.

Additionally, demonstrate that the differential reading h can be magnified by minimising the difference in specific weights ${}_2-{}_1.$ Assume that initially, he fluid levels in the two reservoirs are equal.