Fundamentals of Fluid Mechanics Assignment 1 — Clapeyron, Pascal, and Archimedes  
Question #1
Starting from $\vec{F}=m\vec{a}$ applied on a gas particule, show Clapeyron's ideal gas law: $$ P=\rho R T=N k_{\rm B} T=\frac{R T}{v}$$ 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 $P$ exerts on a hemisphere of radius $R$ is of $P \pi R^2$.
09.03.19
Due on Thursday September 5th at 11:00.
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$\pi$