(Solved): Exercise 6 - Differential pressure gauge [10 points] The figure below shows a differential pressure ...

Exercise 6 - Differential pressure gauge [10 points] The figure below shows a differential pressure gauge. In the first case, both containers are subjected to the same pressure $P_0$. In the second case, $B$ supports a lower pressure $\left(P_0??P\right)$. Both containers have the same cross-section $S$, which is larger than the cross-section $s$ of the connecting tube. 1) Express $?P$ as a function of the densities $?$ and ${?}^{?}$ of the two liquids, the ratio $s/S$ and the height $h$. 2) Under what conditions will the measurement of $?P$ be highly sensitive? 3) Data: $s/S=1/100,?=10^3\mathrm{kg}/{\mathrm{m}}^3$ (water), ${?}^{?}=1.022?10^3\mathrm{kg}/{\mathrm{m}}^3$ (aniline). 4) What is the smallest pressure difference $?P$ that this manometer can measure if the smallest displacement of the interface we can measure is $h=1\mathrm{mm}$ ? Fig. 6. Difterential pressure gauge. Exercise 7 - Densitometer [6 points] A densitometer consists of: a hollow sphere with radius $R=$ $2\mathrm{cm}$ and thickness less than $R$ and a hollow cylindrical rod with outside diameter $d=1\mathrm{cm}$, weighted by a mass. When the densitometer is immersed in water at $15^{?}\mathrm{C}$ and external pressure $P_0$, the sphere is floating the water surface and $x=0\mathrm{cm}$. When immersed in a liquid of density ${?}_l$, the height of the immersed cylinder is $x=8\mathrm{cm}$. What is the density of the liquid? Fig. 7. Densitometer.