(Solved): 21 Figure Q1a shows a manometer consisting of two vertical arms of equal crosssectional area connec ...

21 Figure Q1a shows a manometer consisting of two vertical arms of equal crosssectional area connected to form a U-tube. The cylindrical reservoirs at the top of each arm are identical and of cross-sectional area $A_R$. The fluid in the left arm is water with density ${?}_w$, and the fluid in the right arm is oil with density ${?}_o$ which is less than ${?}_w$. The reservoir on the left is closed with pressurised air. The right reservoir is open to the atmosphere. The position of the oil/water interface is used to measure the difference between the pressures $P_A$ in the left reservoir and atmospheric pressure $\mathrm{B}$ on the right. (i) Find the difference between ${\mathrm{P}}_{\mathrm{A}}$ and $\mathrm{B}$ in terms of measured height of fluid in the U-tube, a, b and c, the density of water, ${?}_w$ and oil ${?}_o$. $[5/25$ marks $]$ (ii) When $P_A=B$, the measured fluid heights are $a=450\mathrm{mm},b=200\mathrm{mm}$ and $\mathrm{c}=300\mathrm{mm}$. Find the relative density of the oil. $[5/25$ marks $]$ (iii) As in (ii) above, when the air pressure above the water is reduced, find the changes to the level of fluid surface $?z$ in the reservoirs as shown in Figure Q1b. Express $?z$ in terms of $H,A_T$ and $A_R$, where $H$ is the drop of the water-oil interface in the $U$-tube, $A_T$ and $A_R$ are the cross-sectional areas of the U-tube and reservoirs, respectively. $[3/25$ marks $]$ (iv) If $P_A$ reduces to $100\mathrm{Pa}$ more than $B$ (i.e. $P_A?B=100\mathrm{Pa}$ ), by how much does $b$ reduce given that ${?}_w=1000\mathrm{kg}/{\mathrm{m}}^3$, relative density of oil is the same as in (ii), $A_R=35A_T$, and $H=0$ when $P_A=B$. $[10/25$ marks] (v) How does the size of the reservoirs at the top of the U-tube affect the sensitivity of the instrument? $[2/25$ marks] Figure Q1a Figure Q1b