1.5 Buoyancy and Archimedes' principle

Definition of buoyancy

Buoyancy is the upward force a fluid exerts on any object placed in it. It's the reason a steel ship floats, a helium balloon rises, and you feel lighter in a swimming pool. Understanding buoyancy connects directly to fluid statics because it emerges from the pressure differences that exist within any fluid at rest.

Upward force

Buoyancy always acts upward, opposing gravity. It arises because fluid pressure increases with depth. The bottom surface of a submerged object sits deeper than the top surface, so it experiences higher pressure. That pressure difference across the object produces a net upward force.

Displaced fluid

When you place an object in a fluid, it pushes some fluid out of the way. The volume of fluid displaced equals the submerged volume of the object. For a fully submerged object, that's the object's entire volume. For a partially submerged object (like a floating block of wood), it's only the volume below the fluid surface.

Magnitude of buoyant force

The buoyant force equals the weight of the displaced fluid, not just its volume. You can calculate it with:

$F_b = \rho_f \, V_{disp} \, g$

• $F_b$ = buoyant force (N)
• $\rho_f$ = density of the fluid (kg/m³)
• $V_{disp}$ = volume of fluid displaced (m³)
• $g$ = acceleration due to gravity (9.81 m/s²)

Note that $V_{disp}$ is the displaced volume, which only equals the object's full volume when the object is completely submerged.

Archimedes' principle

Archimedes' principle states: any object wholly or partially immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces. This single statement is the foundation for all buoyancy analysis, whether the object floats, sinks, or hovers at a fixed depth.

Buoyant force vs. weight of the object

Three outcomes are possible when you place an object in a fluid:

• Object floats ($F_b = W_{object}$): The object displaces just enough fluid for the buoyant force to balance its weight. It settles at the surface with part of its volume above the fluid.
• Object sinks ($F_b < W_{object}$): Even fully submerged, the displaced fluid doesn't weigh enough to support the object.
• Object rises ($F_b > W_{object}$): If you release a light object underwater, the buoyant force exceeds its weight and it accelerates upward until it reaches the surface.

Derivation from hydrostatic pressure

Archimedes' principle follows directly from hydrostatic pressure. Consider a rectangular block submerged in a fluid of density $\rho_f$:

1. The pressure on the top face at depth $h_1$ is $P_{top} = P_0 + \rho_f \, g \, h_1$.

2. The pressure on the bottom face at depth $h_2$ is $P_{bot} = P_0 + \rho_f \, g \, h_2$.

3. The net upward force on the block is $(P_{bot} - P_{top}) \times A = \rho_f \, g \, (h_2 - h_1) \, A$.

4. Since $(h_2 - h_1) \times A$ equals the block's volume $V$, the net upward force is $\rho_f \, g \, V$, which is the weight of the displaced fluid.

This derivation generalizes to any shape through integration of pressure over the object's surface.

Assumptions and limitations

• The fluid is assumed to be incompressible and of uniform density. This works well for most liquids but breaks down in stratified fluids (like ocean layers with varying salinity).
• Surface tension is neglected. For very small objects (needles, insects on water), surface tension can be comparable to or larger than the buoyant force.
• Viscous effects are not included. Archimedes' principle gives the static buoyant force; it doesn't account for drag if the object is moving through the fluid.

Buoyancy calculations

Solving buoyancy problems usually comes down to comparing the buoyant force with the object's weight and applying $F_b = \rho_f \, V_{disp} \, g$. Here's a systematic approach.

Step-by-step method

1. Identify the fluid and find its density $\rho_f$. For fresh water, use 1000 kg/m³; for seawater, roughly 1025 kg/m³.
2. Determine the displaced volume $V_{disp}$. If the object is fully submerged, this equals the object's volume. If it floats, you'll need to solve for $V_{disp}$.
3. Calculate the buoyant force: $F_b = \rho_f \, V_{disp} \, g$.
4. Compare $F_b$ with the object's weight $W = m \, g$ to find the net force and predict whether the object floats, sinks, or is in equilibrium.

Example: floating object

A block of wood with density 600 kg/m³ and volume 0.01 m³ is placed in fresh water ($\rho_f = 1000$ kg/m³). What fraction of the block is submerged?

• At equilibrium, $F_b = W$, so $\rho_f \, V_{disp} \, g = \rho_{obj} \, V_{obj} \, g$.
• Cancel $g$: $V_{disp} / V_{obj} = \rho_{obj} / \rho_f = 600 / 1000 = 0.6$.
• 60% of the block sits below the waterline.

This ratio $\rho_{obj} / \rho_f$ is a quick way to find the submerged fraction of any floating object.

Density of fluid

Fluid density is the single biggest factor controlling buoyant force. Denser fluids push harder. For example, you float more easily in the Dead Sea ($\rho \approx 1240$ kg/m³) than in a freshwater lake ($\rho \approx 1000$ kg/m³) because the denser water displaces more weight per unit volume.

Volume of displaced fluid

For regularly shaped objects, you can calculate volume geometrically. For irregular shapes, submerge the object in a graduated container and measure the rise in fluid level. The volume change equals the displaced volume. This is essentially the method Archimedes himself is said to have used.

Net force on submerged objects

The net force on a submerged object is:

$F_{net} = F_b - W = \rho_f \, V_{disp} \, g - m_{obj} \, g$

• $F_{net} = 0$: object is in equilibrium (neutral buoyancy).
• $F_{net} > 0$: object accelerates upward.
• $F_{net} < 0$: object accelerates downward (sinks).

Stability and equilibrium

A floating object can be in equilibrium (net force and net torque both zero) yet still be stable or unstable depending on how it responds to small disturbances like waves or shifting cargo.

Stable vs. unstable equilibrium

• Stable equilibrium: when tilted slightly, the object experiences a restoring torque that brings it back upright. A wide-hulled boat is a good example.
• Unstable equilibrium: when tilted slightly, the object continues to tip further. Think of a tall, narrow cylinder floating upright with a high center of gravity.
• Neutral equilibrium: the object stays in whatever orientation it's placed in. A uniform sphere submerged at neutral buoyancy behaves this way.

Center of buoyancy (CB)

The center of buoyancy is the point where the buoyant force effectively acts. It's located at the centroid of the displaced fluid volume. When a floating object tilts, the shape of the displaced volume changes, and the center of buoyancy shifts. This shift is what can create a restoring torque.

Center of gravity (CG)

The center of gravity is the point where the object's weight effectively acts. It depends on how mass is distributed throughout the object. For a floating vessel, keeping the CG low (heavy items near the bottom) improves stability.

Metacentric height (GM)

Metacentric height quantifies initial stability for small angles of tilt. It's defined as:

$GM = KB + BM - KG$

where:

• $KB$ = distance from the keel (bottom) to the center of buoyancy
• $BM$ = distance from the center of buoyancy to the metacenter (the point where the line of action of the buoyant force intersects the vessel's centerline when tilted)
• $KG$ = distance from the keel to the center of gravity

Typical cargo ships aim for a GM between 0.15 m and 1.5 m. Too small and the ship is dangerously unstable; too large and it rolls back and forth too sharply, which is uncomfortable and can damage cargo.

Applications of buoyancy

Floating objects

Ships, barges, and buoys all float because their overall density (structure + air inside) is less than the surrounding water. Naval architects carefully calculate the vessel's displacement (the mass of water displaced) to ensure it matches the vessel's total weight under all loading conditions.

Submerged objects

Submarines control their depth by adjusting buoyancy through ballast tanks:

1. To dive, valves open and seawater floods the ballast tanks, increasing the sub's weight until $W > F_b$.
2. To surface, compressed air forces water out of the tanks, decreasing weight until $W < F_b$.
3. To hover at a fixed depth, the sub trims its ballast until $W = F_b$ (neutral buoyancy).

Hydrometers and density measurement

A hydrometer is a simple glass tube weighted at the bottom with a calibrated scale on the stem. When placed in a fluid, it sinks until the buoyant force matches its weight. The denser the fluid, the less the hydrometer sinks, and you read the density directly off the scale. Hydrometers are used to check battery acid concentration, alcohol content in brewing, and salinity in aquariums.

Ballast in ships

Ships use ballast water to maintain stability, especially when sailing without cargo. Filling ballast tanks lowers the center of gravity and increases draft (how deep the hull sits). Proper ballast management prevents capsizing and ensures the propeller and rudder remain sufficiently submerged for effective operation.

Factors affecting buoyancy

Density of object vs. density of fluid

This comparison determines everything:

A solid steel ball ($\rho \approx 7800$ kg/m³) sinks in water, but a steel ship floats because the hull encloses a large volume of air, making the ship's average density much less than water's.

Shape and orientation of object

Shape doesn't change the buoyant force on a fully submerged object (only volume matters), but it strongly affects whether an object floats and how stable it is. A flat hull displaces water over a wide area, creating a high metacentric height. A narrow hull of the same volume is less stable. Orientation also matters: a long cylinder floating on its side is more stable than the same cylinder floating upright.

Compressibility of fluid

Most liquids are nearly incompressible, so their density stays constant with depth. Gases are highly compressible: air density drops significantly with altitude. This means the buoyant force on a balloon decreases as it rises, which is why weather balloons eventually reach a ceiling altitude where $F_b = W$.

Temperature and pressure effects

• Temperature: Fluids generally expand when heated, reducing their density. Warmer water provides slightly less buoyant force than cold water. This effect is small for liquids but significant for gases.
• Pressure: Increased pressure compresses gases and slightly compresses liquids, raising their density. At great ocean depths, water is marginally denser than at the surface, providing a slightly larger buoyant force.

Buoyancy in gases

The same Archimedes' principle applies in gases. The buoyant force on any object surrounded by air is $F_b = \rho_{air} \, V_{obj} \, g$. For most solid objects this force is tiny compared to their weight, which is why you can usually ignore it. But for large, low-density objects, it becomes the dominant force.

Buoyancy in air

At sea level, air density is about 1.225 kg/m³. A 1 m³ object in air experiences a buoyant force of roughly 12 N (about 2.7 lbf). That's negligible for a rock, but for a large balloon filled with a gas lighter than air, it's enough to generate lift.

Hot air balloons

Hot air balloons work by reducing the density of the air inside the envelope:

1. A burner heats the air inside the balloon.
2. The heated air expands. Some spills out the open bottom, reducing the mass of air inside while the volume stays roughly constant.
3. The average density of the balloon (envelope + heated air + basket + passengers) becomes less than the surrounding cooler air.
4. The buoyant force exceeds the total weight, and the balloon rises.

To descend, the pilot allows the air to cool or opens a vent at the top to release hot air.

Archimedes' principle in gases

The formula is identical to the liquid case:

$F_b = \rho_{gas} \, V_{disp} \, g$

For a helium balloon at sea level: $\rho_{air} \approx 1.225$ kg/m³, $\rho_{He} \approx 0.164$ kg/m³. The net lift per cubic meter of helium is roughly $(1.225 - 0.164) \times 9.81 \approx 10.4$ N/m³.

Atmospheric pressure and density

Both pressure and density decrease with altitude. At about 5,500 m, atmospheric pressure is roughly half its sea-level value. A rising balloon experiences progressively less buoyant force. If the balloon is sealed, the gas inside expands as external pressure drops, which can eventually burst the balloon. Weather balloons are designed to expand until they pop at a target altitude, then a parachute returns the instrument package to the ground.

Experimental verification

Measuring buoyant force

The most straightforward lab method:

1. Weigh the object in air using a spring scale or force sensor. Record this as $W_{air}$.
2. Submerge the object fully in the fluid while still attached to the scale. Record the apparent weight $W_{fluid}$.
3. The buoyant force is the difference: $F_b = W_{air} - W_{fluid}$.

You can also measure the volume of fluid displaced (using an overflow can and graduated cylinder) and calculate $F_b = \rho_f \, V_{disp} \, g$ to compare with the direct measurement.

Comparing predicted vs. observed values

Calculate the expected buoyant force from the object's measured volume and the fluid's known density. Then compare it to the experimentally measured value. Agreement within a few percent confirms Archimedes' principle. Larger discrepancies suggest measurement errors or unaccounted factors (trapped air bubbles, for instance).

Sources of error and uncertainty

• Volume measurement: Irregular objects are hard to measure precisely. Air bubbles clinging to the surface increase the apparent displaced volume.
• Fluid density: Temperature fluctuations change fluid density. Even a few degrees can matter if you're aiming for high precision.
• Surface tension: For small objects, surface tension can pull the object down or hold it up, adding a force that isn't part of the buoyancy calculation.
• Instrument calibration: Uncalibrated spring scales or force sensors introduce systematic error.

Improving experimental accuracy

1. Use digital force sensors with resolution of at least 0.01 N.
2. Measure fluid temperature and look up the corresponding density (or measure it with a hydrometer).
3. Tap the submerged object gently to dislodge air bubbles before taking readings.
4. Repeat each measurement at least three times and use the average.
5. Propagate uncertainties through your calculations to report a meaningful error range on your final result.