13.2 Buoyancy and Archimedes' Principle

Archimedes' Principle and Buoyancy

Archimedes' principle tells you exactly how much upward force a fluid exerts on any object placed in it. That upward force, called the buoyant force, equals the weight of the fluid the object displaces. Understanding this single idea lets you predict whether something floats, sinks, or hovers in place.

Physical Basis of Archimedes' Principle

The buoyant force isn't some mysterious property of fluids. It comes directly from the pressure difference between the bottom and top of a submerged object.

Fluid pressure increases with depth because of the weight of fluid stacked above. That means the bottom surface of a submerged object always experiences greater pressure than the top surface. The net result of all that pressure pushing inward from every direction is a force pointing upward.

Archimedes' principle states that this net upward force on a body immersed in a fluid equals the weight of the fluid displaced by that body. It applies whether the object is fully or partially submerged, and it works for any fluid (liquids and gases).

Calculating the Buoyant Force

The buoyant force is given by:

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

where:

• $F_b$ = buoyant force (N)
• $\rho_f$ = density of the fluid (kg/m³)
• $g$ = gravitational acceleration (9.8 m/s² on Earth)
• $V_{\text{disp}}$ = volume of fluid displaced by the object (m³)

Notice that the object's own density doesn't appear in this formula. The buoyant force depends only on the fluid's density and how much fluid volume the object pushes aside.

Three factors control how large the buoyant force is:

• Fluid density. Saltwater ($\rho \approx 1025 \text{ kg/m}^3$) produces a larger buoyant force than freshwater ($\rho \approx 1000 \text{ kg/m}^3$) for the same displaced volume. That's why you float more easily in the ocean.
• Displaced volume. A wide, hollow ship hull displaces a huge volume of water, generating enough buoyant force to support a massive vessel.
• Gravitational field strength. On the Moon ($g \approx 1.6 \text{ m/s}^2$), the same object in the same fluid would experience a smaller buoyant force than on Earth.

Floating and Sinking

Conditions for Floating, Sinking, and Neutral Buoyancy

Whether an object floats or sinks comes down to comparing the object's density to the fluid's density:

• Floats: $\rho_{\text{object}} < \rho_{\text{fluid}}$. The object doesn't need to fully submerge to displace its own weight in fluid, so part of it stays above the surface. Wood ($\rho \approx 500\text{ kg/m}^3$) floats in water because it's about half as dense.
• Sinks: $\rho_{\text{object}} > \rho_{\text{fluid}}$. Even fully submerged, the displaced fluid weighs less than the object, so there's a net downward force. A steel bolt ($\rho \approx 7800\text{ kg/m}^3$) sinks in water.
• Neutrally buoyant: $\rho_{\text{object}} = \rho_{\text{fluid}}$. The buoyant force exactly matches the object's weight, so it neither rises nor sinks. A scuba diver fine-tunes this by adjusting the air in a buoyancy compensator.

For a floating object, you can find what fraction of its volume sits below the surface:

$\frac{V_{\text{submerged}}}{V_{\text{object}}} = \frac{\rho_{\text{object}}}{\rho_{\text{fluid}}}$

Ice has a density of about 917 kg/m³, so in freshwater (1000 kg/m³), roughly 91.7% of an iceberg is underwater. That's where "tip of the iceberg" comes from.

Specific gravity is the ratio of an object's density to the density of water (1000 kg/m³). A specific gravity less than 1 means the object floats in water; greater than 1 means it sinks.

Problem-Solving Strategy

Buoyancy problems follow a consistent approach. Here's how to work through them:

1. Draw a free-body diagram. Identify all forces on the object: its weight ($W = mg$, downward) and the buoyant force ($F_b$, upward). Include any other forces like tension if the object is tethered.

2. Determine the displaced volume. If the object is fully submerged, $V_{\text{disp}} = V_{\text{object}}$. If it's floating, only the submerged portion counts.

3. Apply the buoyant force equation: $F_b = \rho_f \, g \, V_{\text{disp}}$

4. Set up equilibrium or Newton's second law. For a floating or stationary object, $F_b = W$. For a sinking or accelerating object, $F_b - W = ma$.

5. Solve for the unknown. You might need the density formula $\rho = m / V$ to convert between mass, volume, and density.

Example: A block of unknown material has a mass of 2.0 kg and a volume of $2.5 \times 10^{-3} \text{ m}^3$. Does it float in water?

• Object density: $\rho = m/V = 2.0 / (2.5 \times 10^{-3}) = 800 \text{ kg/m}^3$
• Water density: $1000 \text{ kg/m}^3$
• Since 800 < 1000, the block floats.
• Fraction submerged: $800 / 1000 = 0.80$, so 80% of the block sits below the waterline.

Real-World Applications

• Ship design: A steel ship floats because its hollow hull displaces far more water than a solid block of steel would. Naval architects calculate how much cargo a ship can carry before it sits too low in the water.
• Submarines: Ballast tanks fill with water to increase the sub's average density (it sinks) or fill with compressed air to decrease density (it rises). Adjusting the water-to-air ratio allows precise depth control.
• Hot air balloons: Heating the air inside the balloon lowers its density below the surrounding cooler air. The atmosphere exerts a buoyant force on the balloon just as water does on a boat.