8.3 Archimedes' principle

Archimedes' principle explains the upward force that fluids exert on objects placed in them. It's the reason some things float and others sink, and it connects directly to the force and equilibrium concepts you've been building all semester. The core idea is simple: the buoyant force on an object equals the weight of the fluid that object displaces.

Definition of Archimedes' principle

When you place an object in a fluid, the fluid pushes back on it with an upward force. Archimedes' principle tells you exactly how strong that push is: it equals the weight of the fluid the object shoves out of the way.

This single idea explains flotation, sinking, and everything in between. It applies to liquids and gases, which is why it covers both boats on water and balloons in air.

Buoyant force

The buoyant force is the net upward force a fluid exerts on an immersed object. It exists because fluid pressure increases with depth. The bottom of a submerged object sits deeper than the top, so the fluid pushes harder on the bottom surface than it pushes down on the top surface. That pressure difference produces a net upward force.

• Depends on the density of the fluid and the volume of fluid displaced
• Acts on objects in both liquids and gases
• Explains why ships float and hot air balloons rise

Magnitude of buoyant force

The buoyant force equals the weight of the displaced fluid, calculated with:

$F_b = \rho_{fluid} \, g \, V_{displaced}$

where:

• $F_b$ = buoyant force (in Newtons)
• $\rho_{fluid}$ = density of the fluid (in kg/m³)
• $g$ = acceleration due to gravity (9.8 m/s²)
• $V_{displaced}$ = volume of fluid displaced (in m³)

Two things increase the buoyant force: a denser fluid and a larger displaced volume. For a fully submerged object, the displaced volume equals the object's own volume, and the buoyant force stays the same no matter how deep the object goes. Depth doesn't matter because the pressure increase on the top and bottom surfaces is the same.

Direction of buoyant force

The buoyant force always points straight up, opposite to gravity. It originates at the center of buoyancy, which is the centroid (geometric center) of the displaced fluid volume. For floating objects like ships, the relative positions of the center of buoyancy and the center of gravity determine whether the object is stable or tends to tip over.

Fluid displacement

When an object enters a fluid, it pushes fluid aside. The amount of fluid displaced depends on how much of the object is submerged.

Volume of displaced fluid

• Fully submerged object: displaced volume = the object's entire volume
• Partially submerged object: displaced volume = only the volume of the part below the fluid surface

This distinction matters a lot. A floating block of wood only displaces enough water to support its weight, not its full volume.

Weight of displaced fluid

The weight of the displaced fluid is:

$W_{displaced} = \rho_{fluid} \, g \, V_{displaced}$

This quantity is the buoyant force. That's the whole point of Archimedes' principle. Whether an object floats, sinks, or hovers in place depends on how this displaced weight compares to the object's actual weight.

Density and buoyancy

The relationship between an object's density and the fluid's density controls what happens when you put the object in the fluid.

Object density vs. fluid density

There are three possible outcomes:

• $\rho_{object} < \rho_{fluid}$: The object floats. It only partially submerges until it displaces enough fluid to balance its weight. (Example: ice in water, since ice has a density of about 917 kg/m³ vs. water's 1000 kg/m³.)
• $\rho_{object} > \rho_{fluid}$: The object sinks. Even fully submerged, the buoyant force can't match the object's weight. (Example: a steel bolt in water.)
• $\rho_{object} = \rho_{fluid}$: The object is neutrally buoyant and hovers at whatever depth you place it.

Floating vs. sinking objects

A floating object displaces a volume of fluid whose weight equals the object's total weight. That's why a massive steel ship can float: its hull encloses a huge volume of air, making the ship's average density less than water's. The ship sinks just deep enough to displace water weighing as much as the entire ship.

A sinking object displaces its full volume of fluid, but that displaced fluid weighs less than the object, so there's a net downward force.

A neutrally buoyant object (like a properly trimmed submarine) displaces fluid equal to its full volume, and that displaced fluid weighs exactly the same as the object.

Applications of Archimedes' principle

Submarines and ships

• Submarines adjust buoyancy using ballast tanks. Flooding the tanks with seawater increases the sub's average density so it sinks; blowing the water out with compressed air decreases density so it rises.
• Ships are shaped so their hulls enclose large air-filled volumes, keeping average density well below water's density. Naval architects use Archimedes' principle to calculate safe loading limits.
• Life jackets work the same way: low-density foam increases displaced volume relative to weight.

Hot air balloons

Heating the air inside the balloon lowers its density compared to the cooler surrounding air. The atmosphere exerts a buoyant force on the balloon equal to the weight of the displaced outside air. Since the hot air inside weighs less than that displaced cooler air, there's a net upward force. Pilots control altitude by adjusting the burner to change the air temperature inside the envelope.

Hydrometers

A hydrometer is a simple floating instrument that measures liquid density. It's a weighted glass tube with a calibrated stem. In a denser liquid, the hydrometer doesn't need to sink as far to displace its own weight, so more of the stem sticks out. In a less dense liquid, it sinks deeper. You read the density directly off the scale at the fluid surface. Hydrometers are used to check battery acid concentration, antifreeze strength, and alcohol content in brewing.

Mathematical formulation

Buoyant force equation

$F_b = \rho_{fluid} \, g \, V_{displaced}$

This equation tells you that buoyant force scales linearly with both fluid density and displaced volume. If you double the fluid density, you double the buoyant force. If you submerge twice the volume, same thing.

For a fully submerged object with volume $V_{object}$, you can write $F_b = \rho_{fluid} \, g \, V_{object}$ since the entire object is underwater.

Apparent weight calculation

When you hold an object underwater with a spring scale, the scale reads less than the object's true weight. That reading is the apparent weight:

$W_{apparent} = W_{actual} - F_b$

For example, suppose a rock weighs 15 N in air. You submerge it in water and the scale reads 10 N. The buoyant force must be 5 N. From there, you could figure out the rock's volume using $F_b = \rho_{water} \, g \, V$:

$V = \frac{F_b}{\rho_{water} \, g} = \frac{5}{(1000)(9.8)} \approx 5.1 \times 10^{-4} \text{ m}^3$

This is exactly the kind of measurement Archimedes' principle makes possible.

Historical context

Archimedes' discovery

Archimedes of Syracuse (287–212 BCE) is credited with discovering this principle. The famous story says he noticed the bathwater rising as he lowered himself in, and realized the volume of water displaced equaled the volume of his submerged body. That insight connected displaced fluid volume to the upward force he felt.

The golden crown problem

King Hiero II suspected a goldsmith had mixed silver into a crown that was supposed to be pure gold. He asked Archimedes to figure it out without melting the crown down. Archimedes' solution: measure the crown's volume by displacement, then compare its density to that of pure gold. If the crown's density was lower than gold's (19,300 kg/m³), it had been adulterated with a less dense metal. This was one of the first recorded uses of density measurement to detect fraud.

Experimental verification

Buoyancy demonstrations

• Dropping objects of different densities into water shows floating, sinking, and neutral buoyancy directly
• A Cartesian diver (a small sealed container inside a squeezable bottle) shows how pressure changes affect buoyancy: squeezing the bottle compresses the air pocket inside the diver, reducing its volume, increasing its average density, and making it sink
• Helium balloons rising in air demonstrate buoyancy in a gas
• Density columns (layers of honey, water, and oil) let you watch objects settle at different levels depending on their density

Laboratory measurements

• Measure an object's mass and volume to calculate its density, then predict whether it will float or sink
• Use a spring scale or force sensor to measure the buoyant force directly and compare it to $\rho_{fluid} \, g \, V_{displaced}$
• Perform error analysis to see how measurement uncertainty affects your results
• Compare experimental buoyant forces with theoretical predictions to verify the principle

Limitations and assumptions

Ideal fluid conditions

Archimedes' principle, as stated in this course, assumes:

• The fluid has uniform density throughout (no density gradients)
• The fluid is at rest (static equilibrium, no currents)
• Viscosity (internal friction of the fluid) is negligible

In real situations with compressible fluids or strong density gradients (like deep ocean layers), you'd need more advanced analysis.

Effects of surface tension

For very small objects (like a needle "floating" on water or tiny insects walking on a pond), surface tension can support weight in ways Archimedes' principle doesn't account for. The needle's density is higher than water's, so by Archimedes' principle alone it should sink. Surface tension at the air-water interface provides an additional upward force. For most objects at the scale you'll encounter in this course, surface tension effects are negligible.

Real-world considerations

Atmospheric pressure effects

• Changes in atmospheric pressure can shift buoyancy, especially for objects in gases (weather balloons, aircraft)
• At higher altitudes, air density drops, reducing the buoyant force on anything floating in air
• In deep-water diving, pressure compresses air spaces in equipment, changing displaced volumes and buoyancy

Temperature and buoyancy

Fluid density changes with temperature. Warm water is less dense than cold water, so the same object experiences a slightly smaller buoyant force in warm water. This is why submarines and underwater vehicles need to adjust ballast when moving between water layers of different temperatures. Convection currents caused by temperature differences can also create dynamic, non-static conditions where the simple static version of Archimedes' principle is only an approximation.

Problem-solving strategies

Free-body diagrams

Always start buoyancy problems by drawing a free-body diagram. For a submerged or floating object, you'll typically have:

• Weight ($W = mg$) pointing downward
• Buoyant force ($F_b = \rho_{fluid} \, g \, V_{displaced}$) pointing upward
• Any other forces (tension from a string, normal force from a surface, etc.)

Getting these forces on paper makes it much easier to set up your equilibrium or net force equation.

Step-by-step approach

1. List knowns and unknowns. Write down every given quantity and identify what you're solving for.
2. Draw a diagram. Sketch the situation and draw a free-body diagram of the object.
3. Determine the displaced volume. Is the object fully or partially submerged? For floating objects, the displaced volume is not the object's full volume.
4. Write the relevant equations. Typically $F_b = \rho_{fluid} \, g \, V_{displaced}$ and Newton's second law ($\Sigma F = ma$, or $\Sigma F = 0$ for equilibrium).
5. Solve algebraically before plugging in numbers. Isolate the unknown variable first.
6. Substitute values and calculate. Include units at every step.
7. Check your answer. Does the magnitude make sense? Are the units correct? If you calculated that a small rock displaces 500 m³ of water, something went wrong.