Fundamentals of Fluid Mechanics A Assignment 1 — Clapeyron, Pascal, and Archimedes  
Instructions
•  Authorized Equations Only:
- You may only use equations provided in the Tables.
- Derivations or formulas from outside sources are prohibited.
•  Formatting Solutions:
- Use a single-column format.
- Write strictly one statement per line, arranged vertically.
- Do not use multiple equal signs on a single line.
•  Presentation: Solutions must be neat, legible, and easy to verify. Box your final answers.
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•  Grading Policy: Failure to follow these formatting and submission guidelines will result in a lower score and reduced feedback.
Question #1
Starting from $\vec{F}=m\vec{a}$ applied on a gas particule, show Clapeyron's ideal gas law: $$ P=\rho R T=N k_{\rm B} T=\frac{R T}{v}=\frac{\overline{R} T}{\overline{v}}$$ with $$ T\equiv \frac{m \overline{q^2}}{3 k_{\rm B}},~~~~R\equiv \frac{k_{\rm B}}{m}$$ Outline the definition of the pressure $P$, the density $\rho$, the specific volume $v$, and the number density $N$.
07.31.19
Question #2
Starting from the balance of the forces on a fluid at rest prove Pascal's law $\Delta P=-\rho g \Delta y$ with $g$ the gravitational acceleration constant, $\Delta y$ the change in height, and $\Delta P$ the change in pressure. List all assumptions and define clearly all terms used.
Question #3
Starting from the balance of the forces on a fluid at rest prove Archimedes's principle $\vec{F}_{\rm b}=-\rho \vec{g} V$ with $V$ the volume of the displaced fluid, $\vec{g}$ the gravitational acceleration vector and $\rho$ the density of the displaced fluid. List all assumptions and define clearly all terms used.
08.29.19
Question #4
Show through integration that the force that a constant pressure $P_{\rm atm}$ exerts on a hemisphere of radius $R$ is of $P_{\rm atm} \pi R^2$.
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09.03.19
Question #5
Show that the average of a sum of terms is the sum of the averages: $$ \overline{a+b}=\overline{a}+\overline{b} $$
08.27.20
Due on Friday September 4 at 9:00. Do Questions #3 and #4 only.
08.28.26
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