---
title: Fluid Statics
module: Gravitation and Matter
moduleNumber: 6
lessonNumber: 4
order: 604
summary: >
  A fluid at rest cannot support a shear, so the only stress it carries is a
  pressure that must grow with depth to hold up the fluid above it. That single
  balance, $\d p/\d z=-\rho g$, runs the whole subject: it sets manometer readings,
  the force on a dam, and — integrated over a submerged boundary — Archimedes'
  buoyant force $F_B=\rho g V_{\rm disp}$. We derive these, use them to decide when a
  body floats and whether it floats upright, and mark where acceleration, rotation,
  compressibility, or capillarity forces a richer pressure model.
topics: [Gravitation and Matter]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 13 — Fluids; §§13-1–13-3"
---

## Density, pressure, and hydrostatics

Density is mass per volume, $\rho=m/V$. A nonuniform material has local density
$\rho=\d m/\d V$. Specific gravity is density relative to water at a stated
reference temperature and is dimensionless. In a fluid at rest, pressure is the
normal force per area exerted across an imagined surface,

$$
p=\frac{F_\perp}{A}.
$$

Pressure is scalar at a point in a static fluid. If the normal stress differed by
orientation, a small fluid element would experience an unbalanced torque and flow
until isotropy was restored. This property differs from a solid, which can sustain
shear stress. Pressure has SI unit pascal, $\mathrm{Pa}=\mathrm{N\,m^{-2}}$.

Absolute pressure includes atmospheric pressure. Gauge pressure is the excess
above atmosphere, $p_g=p-p_{\rm atm}$. A tyre gauge commonly reports gauge
pressure; thermodynamic equations require absolute pressure. Confusing these two
values produces especially large errors near atmospheric pressure.

**Hydrostatic equation.**

Take a small horizontal fluid slab of area $A$ and height $\d y$, with positive
$y$ upward. The upward force on its lower face minus the downward force on its
upper face equals its weight:

$$
p(y)A-p(y+\d y)A=\rho gA\,\d y,
\qquad \frac{\d p}{\d y}=-\rho g.
$$

For constant density and depth $h$ measured downward from a free surface,

$$
p=p_0+\rho gh.
$$

Container shape does not enter this equation. At one connected horizontal level,
pressure is identical in static fluid regardless of the amount of liquid above
elsewhere in the vessel. The total force on a container wall can nevertheless
depend on shape because wall area and orientation vary.

$$
% caption: A fluid slab is held up by the excess pressure on its lower face over its upper face; the balance $p_bA-p_tA=\rho gA\,\d y$ gives $\d p/\d y=-\rho g$, so pressure rises linearly with depth $h$.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[thick] (0,0)--(5,0);
\node[above right] at (0,0) {free surface};
\draw[black,thick,fill=white] (1.75,-1.3) rectangle (3.25,-2.2);
\draw[->,black,thick] (2.5,-0.6)--(2.5,-1.28) node[midway,right] {$p_tA$};
\draw[->,black,thick] (2.5,-2.95)--(2.5,-2.22) node[midway,right] {$p_bA$};
\draw[->,acc,thick] (2.5,-1.58)--(2.5,-1.98) node[midway,right] {$w$};
\draw[<->] (4.35,0)--(4.35,-1.3) node[midway,right] {$h$};
\end{tikzpicture}
$$

## Buoyancy and Archimedes applications

Archimedes' principle follows from the pressure field without requiring a special
new force law. Imagine that the object is replaced by fluid identical to its
surroundings. The pressure forces exerted on the boundary of that imagined volume
must support the weight of the fluid inside, because the imagined fluid is static.
Removing the imagined fluid and inserting an object leaves the same surrounding
pressure distribution on the same boundary. The net pressure force is therefore
upward and equal in magnitude to the weight of the displaced fluid.

In a uniform-density fluid, this argument gives $F_B=\rho_f gV_{\rm disp}$.
It also explains why the centre of buoyancy is the centre of mass of displaced
fluid: the distributed pressure force has the same resultant line of action that
the displaced fluid's weight would have. A body of irregular shape need not have
a simple geometric centre, but the displaced volume still determines both the
magnitude and line of action of the buoyant force.

The surface-integral description checks signs. Pressure acts
inward normal to the object's surface. Horizontal contributions cancel in a fluid
whose density varies only with height, while vertical contributions add to the
weight of displaced fluid. If density varies with depth, the displaced-fluid
weight must be calculated with the actual density distribution:

$$
F_B=g\int_{V_{\rm disp}}\rho(\vec r)\,\d V.
$$

This form covers a body crossing a density interface. The force still equals the
weight of what would occupy its volume, but a single constant density cannot be
pulled outside the integral. It also identifies the limit of the usual formula:
when the fluid is accelerating, pressure is governed by an effective gravity, and
when surface tension is important, additional forces act at the contact line.

$$
% caption: Pressure acts inward on every face of a submerged body; horizontal contributions cancel while the larger upward push on deeper faces leaves a net buoyant force $F_B=\rho_f gV_{\rm disp}$ equal to the weight of the displaced fluid.
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\draw[->,black,thick] (2.3,-3.15)--(2.3,-2.5);
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\draw[->,black,thick] (3.5,-1.85)--(3.05,-1.85);
\draw[->,acc,very thick] (4.35,-2.7)--(4.35,-1.0) node[midway,right] {$F_B$};
\end{tikzpicture}
$$

**Archimedes applications: hydrometers and ballast.**

A hydrometer is a floating density meter. Its mass and shape are fixed, so its
weight is fixed. In a denser liquid, less volume is needed to displace that
weight, and the instrument floats higher. In a less dense liquid, it sinks deeper.
A narrow stem converts a small change in displaced volume into a measurable change
in immersion depth. The scale is calibrated for a specified temperature because
liquid density and the dimensions of the instrument can both change with
temperature.

Hydrometers illustrate the difference between floating equilibrium and stability.
The displaced volume sets vertical force balance. The bulb shape, ballast, and
centre of gravity determine whether the instrument remains upright. A low mass of
ballast lowers the centre of gravity; a long stem increases the visible change in
draft. Bubbles attached to the bulb or contamination changing the wetting surface
alter the effective displacement and can bias a reading.

Submarines use the same force balance on a larger scale. Flooding ballast tanks
increases the vehicle's mass while changing external displaced volume only slightly,
so the vehicle can become negatively buoyant. Expelling water reduces mass and
restores positive buoyancy. Depth control often uses control surfaces and trim
tanks in addition to main ballast, because neutral buoyancy alone does not set the
orientation or dynamic path of the vehicle.

The phrase "displacement" can refer either to displaced-fluid volume or to a
vehicle's weight expressed as an equivalent fluid mass. A naval displacement value
therefore states mass or weight at a specified draft; hull volume is a different
quantity. Keeping the usage explicit avoids mixing buoyant
volume with an object's own material volume.

## Stratification, stability, and pressure measurement

In a stratified fluid, pressure is continuous across a stable interface but its
gradient changes with density. A sharp density jump occurs, for example, between
fresh water and salt water or between two immiscible laboratory liquids. Starting
from a known surface pressure, move downward through each layer and add
$\rho_i g\Delta h_i$. The final pressure is the sum of the contributions, not the
product of an average density and an arbitrary total depth unless that average has
been defined by the same weighted thicknesses.

Stratification affects buoyancy as well as pressure. A small body crossing an
interface displaces volumes of both fluids. Its buoyant force can change rapidly
as it moves through the interface, and its equilibrium depth may occur where the
combined displaced-fluid weight matches its own weight. This mechanism supports
layered lakes and can trap a neutrally buoyant underwater vehicle near a density
transition.

Pressure gauges connected by tubing require a hydrostatic correction whenever the
gauge port and measurement point are at different elevations or the connecting
line contains a liquid. A remote transducer does not necessarily read the pressure
at a vessel tap without correction. The sign follows from tracing the fluid path:
moving downward to the transducer adds pressure; moving upward subtracts it. A
gas-filled line has a small correction under ordinary conditions, while a liquid-
filled line can have a substantial one.

Gauge selection should match the expected range. A liquid manometer has high
resolution for small differences but limited range. A diaphragm or electronic
transducer can cover a larger range but needs calibration, zero checks, and a
specified reference port. Comparing two gauge types over an overlap range is a
diagnostic when a pressure reading carries significant experimental weight.

$$
% caption: Pressure is continuous across the interface of two immiscible layers ($\rho_2>\rho_1$) but its depth gradient steepens in the denser lower layer, so the pressure-depth graph is a shallow segment above a steeper one.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
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\draw[thick] (0,0)--(2.4,0);
\draw[black,thick] (0,-1.4)--(2.4,-1.4);
\node at (1.2,-0.7) {upper};
\node at (1.2,-2.2) {lower};
\draw[->,black] (3.4,0)--(6.0,0) node[right] {$p$};
\draw[->,black] (3.4,0)--(3.4,-3.0) node[below] {depth};
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\node[black,above right] at (4.35,-1.4) {interface};
\end{tikzpicture}
$$

**Stability curves and operating range.**

The metacentre describes initial stability, but large-angle behaviour is better
described by the righting arm $GZ$. At a given heel angle, the weight and buoyant
force lines are separated horizontally by $GZ$. The righting moment is
$Mg(GZ)$. A positive righting arm tends to restore upright orientation; a negative
one tends to increase heel. A stability curve plots $GZ$ against heel angle and
shows the range over which the vessel has a restoring moment.

The area under a righting-moment curve represents energy required to heel the
vessel through an angle in a quasi-static model. A vessel can have positive initial
metacentric height and still have a limited range of positive stability. Changes in
loading, free-surface liquids, cargo shift, flooding, or a raised centre of gravity
can reduce both the maximum righting arm and the angle at which it vanishes.

Stability curves depend on displacement and hull geometry. They should be computed
or measured for the actual loading condition rather than borrowed from an empty
hull. A high initial slope can feel stiff but may produce rapid, uncomfortable
rolling; a low slope can feel gentle but leave little margin against large heel.
Operational design balances these characteristics with freeboard, reserve buoyancy,
and expected external moments from wind, waves, or turning.

Small laboratory floating bodies can demonstrate the same ideas. Add a movable
mass, record the heel at successive lateral positions, and compare the estimated
righting moment with a geometric model. The procedure must account for the mass
used to tilt the body, the waterline reference, and repeatable initial orientation.
It measures a particular configuration, not an intrinsic property independent of
load and fluid conditions.

$$
% caption: The righting arm $GZ$ rises to a maximum then falls back to zero as heel increases; the vessel has a restoring moment only over the range of positive $GZ$, which ends at the angle of vanishing stability.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(5.8,0) node[right] {heel angle};
\draw[->,black] (0,0)--(0,3.0) node[above] {$GZ$};
\fill[acc!10] (0,0) plot[domain=0:4.7,samples=120] (\x,{2.3*sin(38*\x)*exp(-0.03*\x*\x)}) -- (4.7,0) -- cycle;
\draw[thick] plot[domain=0:4.7,samples=120] (\x,{2.3*sin(38*\x)*exp(-0.03*\x*\x)});
\draw[dashed,black] (4.7,0)--(4.7,0.4);
\node[acc] at (2.5,2.55) {restoring range};
\node[black,above right] at (4.55,0.05) {vanishing};
\end{tikzpicture}
$$

### Interpreting stratification and stability data

Measured pressure and stability curves should be matched to the configuration at
the time of measurement. A density profile sampled at one temperature or salinity
can change after mixing, heating, or evaporation. A gauge correction based on an
assumed liquid-filled line is invalid if the line contains a gas pocket. Simple
checks—recording temperature, inspecting the line for bubbles, and comparing a
known reference pressure—often prevent larger errors than additional numerical
precision would remove.

Stability measurements have similar configuration dependence. A movable test mass
changes both heel moment and total displacement. The water surface must be calm,
the initial orientation repeatable, and any internal liquid level recorded. A
righting-arm curve with sparse angles should not be extrapolated beyond the last
measured point, particularly near deck immersion or an apparent loss of restoring
moment. Report the tested load, fluid, angle range, and uncertainty rather than
treating one curve as an unchanging property of the body.

The constant-density approximation is excellent for liquids over moderate depths.
In an atmosphere, density changes appreciably with pressure and temperature, so the
differential hydrostatic equation must be combined with an equation of state.
With ideal gas at constant temperature, pressure decreases exponentially with
altitude rather than linearly.

**Pressure measurement and Pascal's principle.**

A barometer balances atmospheric pressure against a liquid column. For a mercury
column of height $h$, $p_{\rm atm}=\rho_{Hg}gh$ when the upper space is nearly
vacuum. A U-tube manometer measures a pressure difference from the height
difference of its liquid surfaces. The relevant height is vertical, not distance
along a tilted tube.

Pascal's principle states that a pressure change applied to an enclosed static
fluid is transmitted undiminished throughout the fluid. A hydraulic lift has

$$
\frac{F_1}{A_1}=\frac{F_2}{A_2}.
$$

The output force can exceed the input force, but the input piston travels farther:
$A_1\Delta x_1=A_2\Delta x_2$. Energy conservation remains intact in an ideal
device. Friction, leakage, and fluid compressibility reduce real efficiency.

$$
% caption: An enclosed fluid transmits an applied pressure undiminished, so $p_1=p_2$ forces a force ratio equal to the piston-area ratio $F_2/F_1=A_2/A_1$; the small piston travels farther.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
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\draw[black,thick] (0,-0.5)--(0,-2.2)--(5.7,-2.2)--(5.7,-0.5);
\draw[very thick] (-0.15,-0.75)--(1.5,-0.75);
\draw[very thick] (3.55,-0.75)--(5.85,-0.75);
\draw[->,black,thick] (0.68,0.35)--(0.68,-0.7) node[midway,left] {$F_1$};
\draw[->,black,thick] (4.7,0.55)--(4.7,-0.7) node[midway,right] {$F_2$};
\node[black] at (0.68,-1.9) {$A_1$};
\node[black] at (4.7,-1.9) {$A_2$};
\node[acc] at (2.6,-2.55) {$p_1=p_2$};
\end{tikzpicture}
$$

## Hydrostatic forces and buoyant equilibrium

Pressure increases with depth, so pressure forces on a submerged object do not
cancel vertically. Integrating the pressure over the surface gives an upward
buoyant force equal to the weight of displaced fluid:

$$
F_B=\rho_f gV_{\rm disp}.
$$

This is Archimedes' principle. Horizontal pressure components cancel in a uniform
fluid; the larger upward force on lower surfaces produces the result. The buoyant
force depends on fluid density and displaced volume, not directly on object mass.
An object fully immersed in uniform fluid has constant buoyancy regardless of
depth, provided fluid density is constant and the object does not compress.

A floating object satisfies $F_B=Mg$. Its submerged fraction is therefore
$V_{\rm sub}/V=\rho_{obj}/\rho_f$ for a uniform object. An object sinks if its
average density exceeds fluid density; it rises if lower. A tethered or supported
object need not satisfy this simple equality because tension or support force also
enters vertical force balance.

> **Worked example.** A solid cube of volume $V=0.020\ \mathrm{m^3}$ and density
> $\rho_{\rm obj}=700\ \mathrm{kg\,m^{-3}}$ floats in water
> ($\rho_f=1000\ \mathrm{kg\,m^{-3}}$). What fraction is submerged?
>
> The cube's mass is $M=\rho_{\rm obj}V=(700)(0.020)=14\ \mathrm{kg}$. Floating means
> the buoyant force carries the full weight, $\rho_f gV_{\rm sub}=Mg$, so the displaced
> volume is
>
> $$
> V_{\rm sub}=\frac{M}{\rho_f}=\frac{14}{1000}=0.014\ \mathrm{m^3},
> \qquad
> \frac{V_{\rm sub}}{V}=\frac{\rho_{\rm obj}}{\rho_f}=0.70.
> $$
>
> Seventy percent of the cube sits below the waterline. The fraction is just the
> density ratio, a consequence of force balance rather than any rule that buoyancy
> always equals the object's weight.

**Scope checks.**

Hydrostatic pressure formulas assume a fluid at rest in a frame without significant
acceleration. In an accelerating tank, effective gravity combines real gravity and
the fictitious acceleration. In a rotating fluid, pressure surfaces curve. Surface
tension can matter for small droplets and capillary tubes, where pressure jump
across a curved interface is not negligible. These effects are outside the simple
$p=p_0+\rho gh$ model but do not contradict it within its stated range.

**Hydrostatic force on surfaces.**

Pressure acts normal to every element of a submerged surface. For a horizontal
surface at one depth, pressure is uniform and force is simply $pA$. For a vertical
wall, pressure varies linearly with depth, so the resultant force is the integral

$$
F=\int_A p\,\d A.
$$

With a rectangular wall of width $w$ whose top is at the free surface and whose
height is $H$, gauge-pressure force is $\rho g wH^2/2$. It acts at depth $2H/3$,
below the centroid because lower portions experience greater pressure. A dam must
therefore resist both increasing force and a line of action below midheight.

The atmospheric contribution to pressure acts on both sides of a wall open to air
and cancels in net force calculations. Gauge pressure is convenient for this case.
If one side is vacuum or sealed at a different gas pressure, the absolute pressure
difference must be retained.

**Manometers and communicating vessels.**

At equal depth in one connected static fluid, pressure is equal. This rule is the
most direct way to solve manometer problems. Moving vertically downward through a
fluid adds $\rho g\Delta h$; moving upward subtracts it. At an interface between
immiscible fluids, pressure is continuous even though density changes. A U-tube
with several liquids is solved by tracing one continuous path from known pressure
to unknown pressure and adding each signed hydrostatic change.

An inclined manometer spreads a given vertical height difference over a longer
tube length, making small pressure differences easier to read. Its physics remains
set by vertical height. A barometer's mercury column is high because mercury has
large density; a water barometer at one atmosphere would require a column roughly
$10.3\ \mathrm m$ tall.

> **Worked example.** One side of a mercury U-tube is open to atmosphere and the
> other connects to a gas line; the mercury levels differ by $h=0.120\ \mathrm m$.
> With $\rho_{Hg}=13.6\times10^3\ \mathrm{kg\,m^{-3}}$, find the gauge pressure of the
> gas.
>
> Tracing a static path between the two exposed surfaces, the height difference of the
> connected mercury sets the pressure difference:
>
> $$
> p_g=\rho_{Hg}\,g\,h=(13.6\times10^3)(9.81)(0.120)=1.60\times10^4\ \mathrm{Pa}.
> $$
>
> Only the level difference enters; the height of either column above the bench is
> irrelevant.

## Buoyant equilibrium, apparent weight, and compressibility

The buoyant force acts through the centre of buoyancy, the centre of mass of the
displaced fluid. For a fully submerged symmetric object in uniform-density fluid,
this is the geometric centre of displaced volume. The object weight acts through
its centre of gravity. If these points are not vertically aligned, their forces
form a couple that rotates the body.

With a floating body tilted slightly, the displaced shape changes and the centre of
buoyancy shifts. Stability depends on the relative geometry of the shifted buoyant
force line and the centre of gravity. A restoring couple gives stable floating;
an overturning couple gives unstable floating. Density alone decides whether an
object can float, but shape and centre-of-mass location decide whether it floats
upright.

Ballast lowers a vessel's centre of gravity, increasing resistance to tipping.
It also increases total mass and displaced volume, so practical design balances
draft, freeboard, and stability. A partially filled tank can reduce stability when
fluid shifts laterally; the moving liquid changes the system centre of mass and
can create a continuing overturning moment.

**Apparent weight and measurement.**

A spring scale reading for an immersed object is its apparent weight,

$$
W_{\rm app}=Mg-\rho_f gV_{\rm disp},
$$

when it is held at rest by the scale. The decrease in reading permits density
measurement. If an object weighs $W_{air}$ in air and $W_{water}$ when fully
submerged, then its density is approximately
$\rho_{obj}=\rho_{water}W_{air}/(W_{air}-W_{water})$, after correcting for small
air buoyancy when high precision is needed.

An object resting on the bottom is not necessarily supported only by normal force:
buoyancy remains, but water must contact its surfaces. If a seal excludes water
below the object, the pressure distribution changes and the simple full-displacement
formula cannot be applied without considering the contact region.

**Compressibility and atmospheric pressure.**

Liquids are often approximated as incompressible, but pressure changes do produce
small density changes described by bulk modulus. In deep ocean applications, this
affects density and hydrostatic pressure. For gases, compressibility is essential.
Combining $\d p/\d y=-\rho g$ with $\rho=pM/(RT)$ for isothermal ideal gas gives

$$
p(y)=p(0)\exp\left(-\frac{Mgy}{RT}\right).
$$

Real atmospheric temperature varies with altitude, so the isothermal formula is a
model rather than an exact atmosphere. It nevertheless explains why pressure falls
approximately exponentially over vertical ranges that are large compared with
ordinary laboratory containers.

**Units and common errors.**

Pressure is force per area, not force. The same pressure acting on a larger area
produces a larger net force. Buoyant force is the weight of displaced **fluid**,
not the weight of an arbitrary equal-volume object. A fully submerged object does
not gain buoyant force merely by being lowered deeper in constant-density water.
Every hydrostatics solution should identify whether it uses gauge or absolute
pressure and whether the fluid density is treated as constant.

## Accelerated fluids and surface forces

In a container accelerating horizontally with acceleration $a$, the free surface
tilts until it is perpendicular to the effective gravity vector, the vector sum of
downward gravity and the opposite fictitious acceleration in the container frame.
Its slope satisfies $\tan\theta=a/g$. Pressure still changes in the direction of
effective gravity; it is no longer a function of vertical depth alone. A passenger
watching water in an accelerating vehicle therefore observes a level surface that
is not horizontal in the ground frame.

In a liquid rotating steadily with angular speed $\omega$ about a vertical axis,
the free surface has shape

$$
z=\frac{\omega^2r^2}{2g}+\text{constant}.
$$

Pressure increases outward as well as downward because circular motion requires
inward force. The curved surface is a paraboloid. This result follows from the
same static force balance used in ordinary hydrostatics, now including the
centrifugal term in the rotating frame.

**Capillarity and surface forces.**

At sufficiently small dimensions, surface tension competes with hydrostatic
pressure. A curved liquid surface has pressure difference $\Delta p$ proportional
to surface tension and curvature; for a spherical droplet, $\Delta p=2\gamma/R$.
Wetting and contact angle determine whether a liquid rises or falls in a narrow
tube. These phenomena are excluded from bulk hydrostatic formulas because their
forces scale with perimeter while weight scales with volume.

The distinction sets a modelling rule. Hydrostatic pressure describes the
bulk force distribution in a continuous fluid. Surface tension modifies boundary
conditions at interfaces. A calculation for a millimetre capillary tube needs both;
a calculation for a large reservoir normally needs only hydrostatics.

The appropriate approximation is selected by comparing characteristic pressure
differences. When capillary pressure, compressibility, or acceleration effects are
small relative to $\rho gh$, the ordinary constant-density hydrostatic model gives
reliable forces, buoyancy, and pressure measurements with much less calculation.

## Profiles, calibration, and stability measurements

The hydrostatic equation is local, but engineering questions often require a
resultant force and its line of action. Gauge pressure on a vertical wall grows
linearly from zero at the free surface to $\rho gH$ at depth $H$. The pressure
diagram is triangular. Its area gives the net force per unit width, and its
centroid gives the line of action at two thirds of the depth below the surface.
This location is deeper than the geometric centroid of the wall because the lower
part carries more pressure.

Separating pressure from force avoids a common error. Pressure at the bottom of a
tall narrow vessel can equal pressure at the bottom of a broad vessel when the
fluid depth and density are the same. Their bottom forces differ because the areas
differ. Conversely, two walls of equal area can carry different resultants if one
extends deeper into the liquid. The force calculation must integrate pressure over
the actual submerged geometry.

Layered fluids require a piecewise pressure profile. Pressure remains continuous
at an interface, but its depth slope changes from $\rho_1g$ to $\rho_2g$. A dense
lower layer produces a steeper segment. This is the same rule used in a multi-fluid
manometer: a continuous pressure path is traced through each layer, and every
vertical move contributes a signed $\rho g\Delta h$ for the layer being crossed.

The free-surface pressure also matters. Gauge-pressure calculations set it to
zero only when both relevant external surfaces are exposed to the same atmosphere.
A sealed tank with gas pressure above the liquid has a rectangular pressure
contribution from that gas in addition to the triangular liquid contribution.
The resultant then moves upward relative to the pure triangular case even though
the liquid's pressure gradient is unchanged.

$$
% caption: Gauge pressure on a vertical wall grows linearly from zero at the surface to $\rho gH$ at the base; the triangular load puts the resultant at depth $2H/3$, below the wall's midheight.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
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\draw[black,very thick] (0,0.2)--(0,-3.0);
\draw[thick] (0,0)--(2.4,-3.0);
\foreach \y/\len in {-0.5/0.4,-1.0/0.8,-1.5/1.2,-2.5/2.0}{
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\draw[<->,black] (-0.35,0)--(-0.35,-3.0) node[midway,left] {$H$};
\end{tikzpicture}
$$

**Manometers, calibration, and pressure readings.**

A manometer is a primary hydrostatic comparator: it converts a pressure
difference into a height difference. Its accuracy depends on the liquid density,
the vertical height reading, the local value of gravity when high precision is
needed, temperature, and the geometry used to read the meniscus. The liquid does
not need to be mercury; a lower-density liquid gives a larger height difference
for the same pressure difference and can improve resolution for small pressure
changes, at the cost of a taller instrument.

The reference side of a U-tube must be identified. If it is open to atmosphere,
the result is gauge pressure. If it is connected to a sealed reference chamber,
the result is a pressure difference from that chamber. A pressure transducer can
be calibrated against a manometer by applying several stable pressures and fitting
its output against the hydrostatic reference. The calibration range, direction of
pressure change, and temperature should be recorded because mechanical sensors can
show zero drift or hysteresis.

Meniscus reading is a practical uncertainty source. For a wetting liquid in a
glass tube, the conventional reading is the bottom of the concave meniscus at eye
level. Parallax is reduced by viewing perpendicular to the scale. In an inclined
manometer, the long displacement along the tube is not itself used in
$\Delta p=\rho g\Delta h$; it must be converted to its vertical component. The
greater tube length improves reading resolution without changing the hydrostatic
law.

When density varies appreciably with temperature, use the density appropriate to
the measured temperature or include the resulting uncertainty. A reading taken
while the fluid is oscillating is not a static measurement. Wait for the levels to
settle, then record both levels or their difference together with the reference
pressure condition.

$$
% caption: A U-tube manometer reads a gas pressure from the vertical height difference $h$ of its two liquid surfaces; an open reference side makes the reading a gauge pressure.
\begin{tikzpicture}[>=Latex,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,thick] (0.7,2.7)--(0.7,0.6)--(2.9,0.6)--(2.9,2.7);
\draw[black,thick] (1.2,2.7)--(1.2,1.0)--(2.4,1.0)--(2.4,2.7);
\draw[dashed,black] (0.7,2.15)--(3.45,2.15);
\draw[dashed,black] (2.9,2.55)--(3.45,2.55);
\draw[<->,acc,thick] (3.3,2.15)--(3.3,2.55) node[midway,right] {$h$};
\draw[->,black,thick] (0.95,3.15)--(0.95,2.75);
\node[black,above] at (0.95,3.15) {$p$};
\node[black,above] at (2.65,2.7) {open};
\end{tikzpicture}
$$

**Floating stability and metacentres.**

In upright floating equilibrium, weight acts downward through the
centre of gravity $G$ and buoyancy acts upward through the centre of buoyancy $B$.
After a small heel, the displaced volume changes shape and $B$ moves to the low
side. The new buoyant-force line intersects the original vertical centreline at a
point called the metacentre $M$ for small rotations. If $M$ lies above $G$, the
separated weight and buoyancy lines form a restoring couple. If $M$ lies below
$G$, the couple increases the heel and the upright orientation is unstable.

The distance $GM$ is the metacentric height. It is a local small-angle stability
measure, not a complete guarantee against capsize at large heel. Hull shape can
produce a changing righting arm as angle increases, and deck immersion or flooding
can alter the displaced volume abruptly. Nonetheless, metacentric height explains
why a low centre of gravity and sufficient waterplane breadth usually improve
initial stability.

Free surfaces inside a vessel can reduce stability. When liquid in a partially
filled tank moves to the low side during heel, its centre of mass shifts and adds
an overturning contribution. Baffles reduce this motion; ballast placed low lowers
the overall centre of gravity. Both changes must be assessed with mass, buoyancy,
and available volume rather than by density alone.

A fully submerged body retains fixed uniform-fluid displacement under a small
rotation. Rotational stability depends directly on the relative positions of its
fixed centre of buoyancy and centre of gravity. A submarine may use ballast
distribution to place its centre of gravity below its centre of buoyancy, giving a
restoring couple.

$$
% caption: Heeling a floating body moves the centre of buoyancy $B$ to the low side; the new buoyant line meets the hull centreline at the metacentre $M$. With $M$ above the centre of gravity $G$, the offset weight and buoyancy form a restoring couple.
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\draw[black,thick,fill=white] (1.6,0)--(4.6,0)--(4.15,0.85)--(2.05,1.0)--cycle;
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\draw[->,black,thick] (3.42,0.5)--(3.42,2.05);
\node[black,right] at (3.42,1.35) {buoyancy};
\draw[->,thick] (3.1,0.55)--(3.1,-0.55) node[below] {weight};
\draw[fill=black!55,draw=black] (3.1,0.55) circle (1.7pt) node[left] {$G$};
\draw[fill=white,draw=black] (3.42,0.28) circle (1.7pt) node[right] {$B$};
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$$

**Fluid-statics measurement and model checks.**

Fluid statics experiments are most reliable when the measured quantity and the
assumed model are kept distinct. A pressure probe measures a local pressure
difference relative to its reference port. A manometer measures a height
difference. A spring balance measures tension, from which buoyancy is inferred
only after a free-body diagram is written. Recording all raw quantities permits an
independent residual test of the hydrostatic relation.

A pressure-versus-depth experiment should use several depths, allow the fluid to
settle after moving the probe, and fit a line. The slope estimates $\rho g$ and
the intercept estimates the reference pressure. A curved trend can indicate a
changing density, acceleration, trapped gas, sensor nonlinearity, or a depth
reference error. Repeating points during upward and downward motion tests for
probe hysteresis and meniscus or sensor lag.

Buoyancy measurements benefit from taring the scale, measuring displaced volume
independently when possible, and checking that the object is fully wetted without
touching the container. Surface tension at a suspension wire, bubbles attached to
the object, and liquid adhering during removal can each change the apparent force.
A floating object's draft should be referenced to a stable free surface and repeated
around the perimeter when the body may tilt.

The uncertainty budget should include depth resolution, fluid-density uncertainty,
temperature, force-scale calibration, geometry, and the validity of the
constant-density approximation. The final graph should show units, error bars or
their stated absence, the fitted model, and residuals. Agreement with
$p=p_0+\rho gh$ over a stated range supports the hydrostatic model in that range;
it does not validate it for a moving, compressible, or capillary-dominated fluid.

**Hydrostatic weighing and density inference.**

Buoyancy determines density from two force readings of the same
object. Let $W_a$ be the weight measured in air and $W_f$ the
apparent weight when the object is fully submerged in a fluid of known density.
Neglecting the small buoyancy of air, the difference is the buoyant force:

$$
W_a-W_f=\rho_f gV.
$$

The object density follows from its weight and displaced volume. This approach is
well suited to an irregular solid whose volume is difficult to measure from
dimensions. It requires complete immersion, no contact with the vessel, and no
trapped bubbles. A porous or absorbent specimen needs additional care because the
fluid can enter its pores and change both mass and displaced volume during the
measurement.

The force difference can be small for a dense object in a low-density fluid, so
scale resolution and calibration matter. The two readings share the same object
mass and often the same balance zero, creating correlated uncertainty. Taking a
difference cancels some common offset but does not cancel a force-scale error that
changes the reading's gain. Repeating the immersion with a known reference object
tests the full procedure, including scale, suspension, wetting, and volume
assumptions.

Hydrostatic weighing also clarifies apparent weight in ordinary problems. A
submerged object pulled downward by a tether can have buoyancy larger than its
weight; the tether supplies the additional downward force. An object held upward
by a scale has buoyancy smaller than its weight; the scale supplies the remaining
upward force. Archimedes' principle sets the pressure-force contribution, while
the free-body diagram determines the other forces required for equilibrium.

**Draft, loading, and Archimedes examples.**

In a floating vessel, an added mass $\Delta m$ requires an added displaced volume
$\Delta V=\Delta m/\rho_f$. Near one operating draft, the change in draft is
approximately $\Delta T=\Delta V/A_w$, where $A_w$ is the waterplane area. A wide
waterplane produces a small change in draft for a given added mass; a narrow one
produces a larger change. This local relation is an approximation because the
waterplane area can change as the hull sinks.

Load lines and freeboard are direct applications. Freeboard is the vertical
distance from the operating waterline to a deck or opening. Adding cargo lowers
freeboard even when the vessel remains buoyant. A vessel may carry its weight
without sinking yet be unsafe in waves if reduced freeboard allows water ingress
or shifts the effective loading condition. Buoyant force balance is necessary but
does not capture all operational limits.

A floating pontoon supplies a simple numerical model. If its vertical sides have
constant plan area $A$, the submerged depth is $T=M/(\rho_f A)$. Doubling the
load doubles the draft until the geometry changes or the deck reaches water. This
is not the same relation for a tapered hull, where the displaced volume must be
computed from the actual shape. Static diagrams should distinguish a geometric
draft calculation from a prediction of stability under a changing load.

**Neutral levels in layered liquids.**

A body placed in a density gradient may settle at a neutral level between the free
surface and the bottom. At each depth, its buoyancy equals the weight of the local
displaced fluid. If the body is denser than the upper layer
but less dense than the lower layer, it can rest with part of its volume in each
layer. The displaced volumes adjust until the combined buoyant force equals its
weight. This is a static equilibrium of the same Archimedes law applied piece by
piece.

The equilibrium can be stable. Raising the body may replace dense displaced fluid
with lighter fluid, reducing buoyancy below weight and pulling it downward.
Lowering it can replace light displaced fluid with denser fluid, increasing
buoyancy and pushing it upward. The argument is a buoyancy version of a potential-
energy minimum. It is relevant to density-stratified water and to instruments that
are designed to settle at a chosen density layer.

Mixing changes the model. A body moving quickly can stir an interface or entrain
one fluid into the other, so the assumed sharp density boundary no longer remains
static. Temperature gradients can also create a continuous density profile, in
which case the restoring force is related to the local density gradient rather
than a discrete jump. The static picture is reliable only after the fluid has
settled and density variation is known.

### Practical limits on buoyancy tests

The simplest buoyancy calculation assumes a quiescent fluid, a known density, and
an object whose shape and volume do not change under pressure. Flexible containers,
foams, gas-filled bodies, and porous materials can violate one or more of these
assumptions. A compressible body may displace less volume at greater depth; a
porous body may absorb liquid; a gas bubble may change volume enough to alter the
force during the measurement. The appropriate response is to state the model
limit, measure the relevant change when it matters, and avoid presenting one
constant-density result as though it applied across every depth and loading state.

In a laboratory tank, wall proximity and a shallow bottom can also alter the
idealized flow-free setting when a body is moved or when its displaced volume is a
substantial fraction of the tank. Waiting for the water to settle restores static
conditions, but it does not remove geometric constraints. A reliable buoyancy
report records fluid type, temperature, immersion condition, contact checks, and
the range over which the static model was tested.

**Measuring a stability curve.**

A small-scale stability experiment can estimate a righting curve by applying a
known lateral moment and measuring the resulting heel. If a test mass $m_t$ is
moved a known horizontal distance $x$ on a floating model, its applied heel moment
is approximately $m_tgx$ for small angles. At equilibrium this is balanced by the
hydrostatic righting moment. Repeating the measurement over several positions
gives points on a curve of righting moment or righting arm against heel angle.

The model must be allowed to settle after each adjustment. Water sloshing, friction
at a heel indicator, and a test mass that changes the total displacement all affect
the result. A camera or plumb line can measure heel angle, but its reference must
be vertical rather than the laboratory bench if the tank is not level. Reversing
the sequence of applied moments tests for hysteresis from liquid shift, support
friction, or a changing internal configuration.

The measured curve should be reported with the total mass, draft, liquid density,
test-mass geometry, and angle convention. Small-angle data establish only initial
stability. Approaching large heel can change waterline shape, immerse openings, or
move liquid cargo; those effects require a separate model or direct observation.
The purpose of the curve is to state the restoring range for one configuration,
not to attach an unconditional stability label to every possible loading state.