Fluid Mechanics Assignment 1 — Fluid Statics  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
Consider the following inclined-tube reservoir manometer:
Find the pressure inside the tank $P_0$ knowing $P_1=1~$atm, $A_0=1~{\rm m}^2$, $A_1=0.1~{\rm m}^2$, $\rho=1000$ kg/m$^3$, $L=1.0~$m, $h=0.1$ m, and $\alpha=(30+\xi_1)^\circ$. Does the sensitivity of the manometer increase with increasing angle $\alpha$? Note that $A_0$ is the liquid surface in the reservoir, $A_1$ is the area of the tube (not the area of the liquid interface).
Question #2
Consider the following embankment made of concrete, with the shape of a right-angled triangle with the base $L$ and the height $h$:
Determine the lowest friction coefficient between the concrete and the ground that will prevent the concrete embankment from sliding. Take $L=(2+\xi_1)$ m, $h=1$ m, $\rho_{\rm w}=1000$ kg/m$^3$ and $\rho_{\rm c}=2400$ kg/m$^3$.
Question #3
Consider the following embankment made of concrete, with the shape of a right-angled triangle with the base $L$ and the height $h$, and fixed to the ground at point A:
Determine the minimum length $L$ the concrete embankment must have to avoid tipping over point A. Take $h=1$ m, $\rho_{\rm w}=1000$ kg/m$^3$ and $\rho_{\rm c}=(2400-9\times\xi_2)$ kg/m$^3$.
Question #4
Consider a U-tube manometer mounted on a trolley which is undergoing an acceleration $\vec{a}$:
What is the magnitude of the acceleration if $L=10$ cm, $h=5$ cm, and $\rho=1000$ kg/m$^3$?
Question #5
Consider a manometer formed from glass tubing with a uniform internal diameter $D$:
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The U-tube is initially partially filled with water at a density of 1000 kg/m$^3$. Then, 3.25 cm$^3$ of olive oil with a density of 800 ${\rm kg/m}^3$ is added to the left side. If the internal diameter $D$ is $(6.35+\xi_1)$ mm, Calculate the equilibrium height, $H$, when both ends of the U-tube are open to the atmosphere.
Question #6
A channel with a height $h$ and depth $d$ is closed by a plate, as follows:
The plate is hinged at point A. To open the channel, a mass $m$ is moved a certain distance $L$ from the water level as indicated on the figure. For $h=1~$m, $d=1$ m, $m=100$ kg, $\alpha=(10+0.5\times\xi_2)^\circ$, and $\rho=1000$ kg/m$^3$, determine the minimum length $L$ needed to open the channel.
Question #7
A float is used to keep constant the level of fuel in a carburetor:
The float has a diameter $D$ and length $L$, and contains a gas with a density $\rho_{\rm g}$. The motion of the float is restricted, effectively preventing the height of the top surface of the float to exceed $h_{\rm stop}$. For $L=10~$cm, $D=(5+\xi_1)$ cm, $\rho_{\rm f}=900$ kg/m$^3$, $\rho_{\rm g}=1$ kg/m$^3$, $h_{\rm stop}=20$ cm, plot the force acting on the needle as a function of the fuel height $h_{\rm f}$ (consider the fuel height range $0\le h_{\rm f} \le H$). The weight of the brass shell encasing the float can be neglected. Use Pascal's law (and not Archimedes' buoyancy principle) to solve this problem.
Due on Wednesday September 24th
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