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The
absolute pressure
$P$ of a fluid of density $\rho$
at a depth $h$ from its surface that is exposed to an
atmospheric pressure $P_0.$ This equation says that the
absolute pressure applied to the surface at depth $h$ is
the sum of the pressure applied at the surface (depth 0)
plus the pressure applied by the weight of the volume
above depth $h.$ It is also the magnitude of the reaction
force (per unit area, i.e. pressure) acting upward on the
volume of water above the depth $h$ by Newton’s third law.
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\[
P=P_0+\rho gh
\]
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gauge pressure.
This is the pressure of the column
of water with depth $h$ resulting from
the water’s weight alone (not including
pressure on the water from surrounding
water or air media).
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\[
P_{\mathrm{gauge}}=P-P_0=\rho gh
\]
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weight of a rectangular cylinder
of density $\rho,$ mass $m,$ cross sectional area $A,$ and height $h.$
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\[
F_{\mathrm{gauge}}=P_{\mathrm{gauge}}A=\rho ghA=\rho gV=mg
\]
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weight-density (weight per unit volume)
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\[
\rho g=mg/V
\]
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weight in terms of density
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\[
\rho gV=mg
\]
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open-tube manometer
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mercury barometer
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\[
\begin{align*}
0 &=P=P_0-\rho gh\\
P_0 &=\rho gh
\end{align*}
\]
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mercury barometer,
cf. p. 16
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\[
P=P_0+\Delta P_0
\]
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units, pressure
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\[
\left[\rho gh\right]
=\frac{\mathrm{M} }{\mathrm{L}^3}\frac{\mathrm{L} }{\mathrm{T}^2}\mathrm{L}
=\frac{1}{\mathrm{L}^2}\frac{\mathrm{ML}}{\mathrm{T}^2}
=\frac{\mathrm{F}}{\mathrm{L}^2}
\]
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atmospheric pressure
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\[
P_0=1\cdot\mathrm{atm}=1.013\times{10}^5\cdot\mathrm{Pa}
\]
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equation of continuity
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\[
A_1v_1=A_2v_2=\mathrm{constant}
\]
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Archimedes’ Principle
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\[
F_b=F_{g,\mathrm{fluid}}
\]
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Bernoulli’s Equation
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\[
P+\frac{1}{2}\rho v^2+pgy=\mathrm{constant}
\]
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Alex Notes