$($a$)$ Cavitation is a problem that occurs in pumps where there is insufficient Net Positive Suction Head on the suction side of the pump. If a cavitating pump produces vapour bubbles of $0\mathrm{.}01$ mm diameter, calculate the pressure of these bubbles impinging on the pump's impeller. Assume that the surface tension coefficient is $0\mathrm{.}072$N$/$m$.$

$($b$)$ From first principles, show that the gauge pressure of a gas inside a vessel measured by a manometer can be given by the following equation:

Where

$P=g({}_2{h}_2-{}_1{h}_1)$

${}_2$ is the density of the manometer fluid

${}_1$ is the density of the gas inside the vessel

$h_1$ is the vertical distance from the vessel to the manometer datum line

$h_2$ is the vertical distance from the manometer datum line to the level of liquid in the right hand limb of the manometer.

Why is water never used as a manometer fluid?

$($c$)$ Explain, in terms of intermolecular$/$interatomic forces, why water forms irregular smears on glass whereas mercury forms droplets of circular cross$-$section.