11.7 Archimedes’ Principle

When you're in a pool, you feel lighter. That's buoyancy at work. Archimedes' principle explains this upward force: it equals the weight of fluid an object displaces. This single idea governs why boats float, why submarines control their depth, and why some objects sink straight to the bottom.

Whether something floats or sinks comes down to density. If an object is less dense than the surrounding fluid, it floats. If it's more dense, it sinks. You'll see this principle applied in ship design, hot air balloons, and even scuba diving.

Archimedes' Principle and Buoyancy

Buoyant force and Archimedes' principle

Any object immersed in a fluid experiences an upward push called the buoyant force. This force counteracts gravity, which is why objects feel lighter underwater than they do in air. The difference between an object's actual weight and how heavy it feels submerged is sometimes called its apparent weight.

Archimedes' principle puts a number on that upward push: the buoyant force equals the weight of the fluid the object displaces. If you lower a block into water and it pushes aside 0.002 m³ of water, the buoyant force equals the weight of that 0.002 m³ of water.

The formula is:

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

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

Note that $V_{displaced}$ equals the submerged volume of the object, not necessarily its total volume. A floating object only displaces a fraction of its volume.

Density's role in floating vs sinking

The comparison between an object's average density and the fluid's density determines what happens:

• Object less dense than fluid (e.g., wood in water): The buoyant force can exceed the object's weight, so the object rises. It keeps rising until only enough volume is submerged to displace a weight of fluid equal to the object's weight. At that point, the object floats in equilibrium.
• Object more dense than fluid (e.g., a rock in water): The object's weight exceeds the maximum buoyant force (even when fully submerged), so it sinks.
• Object with the same density as the fluid: The buoyant force exactly equals the object's weight at any depth. The object is neutrally buoyant and will hover wherever you place it. Submarines aim for this condition when they want to hold a steady depth.

Calculations with Archimedes' principle

Here's a step-by-step approach for solving buoyancy problems:

1. Find the displaced volume. For a fully submerged object, $V_{displaced}$ equals the object's total volume. For a floating object, it's only the portion below the fluid surface.

   • For regular shapes (cubes, spheres, cylinders), calculate volume with standard geometry formulas.
   • For irregular shapes, submerge the object in a graduated container and measure how much the fluid level rises.

2. Identify the fluid density. Common values: freshwater is about 1000 kg/m³, seawater is about 1025 kg/m³.

3. Apply the formula: $F_b = \rho_{fluid} \times V_{displaced} \times g$

4. Compare forces. If you need to know whether the object floats or sinks, compare $F_b$ to the object's weight $W = mg$. The net force is $F_b - W$: positive means it rises, negative means it sinks, zero means equilibrium.

Quick example: A 0.005 m³ block is fully submerged in freshwater.

$F_b = 1000 \, \text{kg/m}^3 \times 0.005 \, \text{m}^3 \times 9.8 \, \text{m/s}^2 = 49 \, \text{N}$

If the block weighs 30 N, the net upward force is 19 N, so it rises until it floats.

Fluid density effects on buoyancy

Because $\rho_{fluid}$ sits right in the buoyant force equation, changing the fluid changes the force.

• Saltwater vs. freshwater: Seawater (≈1025 kg/m³) provides a stronger buoyant force than freshwater (≈1000 kg/m³). That's why you float noticeably higher in the ocean, and ships ride higher in saltwater ports.
• Temperature effects: Warming a fluid generally lowers its density. Hot air balloons work on this idea: the burner heats the air inside the envelope, making it less dense than the cooler air outside, so the balloon experiences a net upward force.

Common buoyancy applications:

• Ships and submarines use hull shape and ballast tanks to control how much water they displace and therefore how they float or dive.
• Scuba divers adjust air in a buoyancy compensator device (BCD) to fine-tune whether they rise, sink, or hover.
• Airships and weather balloons use low-density gases like helium to displace heavier surrounding air, generating lift.

Factors affecting buoyancy

Several variables work together to determine the net force on a submerged object:

• Volume of the object controls how much fluid gets displaced. A larger object displaces more fluid and experiences a greater buoyant force.
• Fluid density sets how heavy each unit of displaced volume is. Denser fluids produce stronger buoyant forces.
• Weight of the object ($W = mg$) acts downward, opposing the buoyant force. The net force on the object is $F_{net} = F_b - W$.
• Depth does not change the buoyant force on a fully submerged object of fixed volume. Hydrostatic pressure increases with depth, but it increases equally on the top and bottom of the object, so the pressure difference (and therefore the buoyant force) stays the same. This is a common point of confusion on exams.