Fundamentals of Fluid Mechanics A Assignment 2 — Forces in Fluids at Rest  
Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Failure to do this will result in a lower score and fewer comments on my part.
Question #1
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 $h=2$ m, $\rho_{\rm w}=1000$ kg/m$^3$ and $\rho_{\rm c}=2400$ kg/m$^3$.
Question #2
(a)  The dirigible Hindenberg had a gas capacity of 200,000 cubic meter. Compare the lifting force at 1 atm and 21$^\circ$C when filled with hydrogen and when filled with helium. (Remember the principle of Archimedes — that buoyancy is equal to the weight of the displaced fluid).
(b)  The Hindenberg could ascend to a maximum altitude of 7,620 meters when filled with hydrogen. The air temperature at this altitude is -18$^\circ$C and the pressure is 1/3 atm; assume that the hydrogen temperature and pressure also have these values. Estimate the weight of the Hindenberg if it were filled with air at 1 atm and placed on a set of scales at sea level.
Question #3
A channel with a height $h$ and depth $d$ is closed by a plate, as follows:
figure3.png  ./download/file.php?id=5356&sid=b43f2947d8978a98227ac4b667ed9c15  ./download/file.php?id=5356&t=1&sid=b43f2947d8978a98227ac4b667ed9c15
The plate is hinged to the ground as indicated in the figure. To keep the channel closed, a mass $m$ is moved a certain distance $L$ from the hinge. For $h=1~$m, $d=1$ m, $L=2$ m, $\alpha=35^\circ$, and $\rho=1000$ kg/m$^3$, determine the mass $m$ such that the plate stands in equilibrium as shown in the figure.
Question #4
A canoe is represented by a right circular semicylinder, with $R=0.35$ m and $L=5.25$ m. The canoe floats in a pool of water of depth $d$ with the water density being $\rho=1000$ kg/m$^3$.
(a)  Using Archimedes' principle, set up a general algebraic expression for the maximum total mass (canoe and contents) that can be floated as a function of the depth $d$.
(b)  Repeat part (a) but without using Archimedes' buoyancy principle. Rather, do so by integrating the forces on the canoe using Pascal's law.
(c)  Using the algrebraic expression derived in (b) or (c), find the total mass (canoe and contents) that can be floated when $d=0.245$ m.
Question #6
Consider mercury in a tank in which a U-shaped capillary tube is inserted as follows:
problem6.png  ./download/file.php?id=5358&sid=b43f2947d8978a98227ac4b667ed9c15  ./download/file.php?id=5358&t=1&sid=b43f2947d8978a98227ac4b667ed9c15
Knowing that $H_1=1~$m, $H_2=0.95~$m, $D_1=0.002~$m, $D_2=0.005~$m, that the wetting angle $\phi=129^\circ$, that the surface tension for mercury is of $\sigma=0.466$ N/m, and the density of mercury is of 13600 kg/m$^3$, do the following:
(a)  Find the height $H_3$.
(b)  Find the air pressure within the capillary tube, $P_{\rm B}$.
(c)  Find the air density within the capillary tube, $\rho_{\rm B}$.
Hint: assume that the system is at steady-state and that there is no change in time of any fluid property.
1.  1.49 m.
3.  309 kg.
4.  630.2 kg.
6.  1.2516 kg/m$^3$.
Due on September 17th at 11am. Do Problems #3, #4, and #6 only.
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