Buoyancy

Overview

Buoyancy is the upward force exerted by a fluid on an immersed or floating object. This force allows ships to float and hot air balloons to rise.

Archimedes' Principle

An object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced.

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

Where:

• $F_b$ = buoyant force
• $\rho_{\text{fluid}}$ = density of the fluid
• $V_{\text{displaced}}$ = volume of fluid displaced
• $g$ = gravitational acceleration

Floating and Sinking

Condition for Floating

Object floats when:

$$F_b \geq W_{\text{object}}$$

Or equivalently:

$$\rho_{\text{object}} \leq \rho_{\text{fluid}}$$

Condition for Sinking

Object sinks when:

$$\rho_{\text{object}} > \rho_{\text{fluid}}$$

Neutral Buoyancy

Object hovers when:

$$\rho_{\text{object}} = \rho_{\text{fluid}}$$

Floating Objects

For a floating object in equilibrium:

$$F_b = W$$ $$\rho_{\text{fluid}} \times V_{\text{submerged}} \times g = \rho_{\text{object}} \times V_{\text{total}} \times g$$

Fraction Submerged

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

Fraction Above Surface

$$\frac{V_{\text{above}}}{V_{\text{total}}} = 1 - \frac{\rho_{\text{object}}}{\rho_{\text{fluid}}}$$

Apparent Weight

The apparent weight of an object in a fluid:

$$W_{\text{apparent}} = W - F_b = \rho_{\text{object}} \times V \times g - \rho_{\text{fluid}} \times V \times g$$ $$W_{\text{apparent}} = V \times g \times (\rho_{\text{object}} - \rho_{\text{fluid}})$$

In Terms of True Weight

$$W_{\text{apparent}} = W\left(1 - \frac{\rho_{\text{fluid}}}{\rho_{\text{object}}}\right)$$

Density Comparison

Material	Density (kg/m³)
Air	1.29
Ice	917
Water	1000
Seawater	1025
Iron	7870
Mercury	13,600
Wood (oak)	750
Cork	240
Aluminum	2700

Stability of Floating Objects

Center of Buoyancy (B)

Centroid of the displaced fluid volume

Center of Gravity (G)

Where the weight acts

Metacenter (M)

Point where the vertical through the new center of buoyancy intersects the original vertical

Stability Conditions

• M above G: Stable (righting moment)
• M below G: Unstable (capsizing moment)
• M at G: Neutral

Examples

Example 1: Buoyant Force

A 0.5 m³ block is fully submerged in water.

$$F_b = \rho V g = 1000 \times 0.5 \times 9.8 = 4900 \text{ N}$$

Example 2: Floating Ice

An ice cube ($\rho = 917$ kg/m³) floats in water. What fraction is submerged?

$$\text{Fraction submerged} = \frac{\rho_{\text{ice}}}{\rho_{\text{water}}} = \frac{917}{1000} = 0.917 = 91.7\%$$

About 8.3% is above water (tip of the iceberg!)

Example 3: Floating Wood

A log ($\rho = 600$ kg/m³, volume = 0.1 m³) floats in water. Find submerged volume.

$$V_{\text{sub}} = V \times \frac{\rho_{\text{wood}}}{\rho_{\text{water}}} = 0.1 \times \frac{600}{1000} = 0.06 \text{ m}^3$$

Example 4: Ship Loading

A ship has waterline area of 2000 m² and displaces 20,000 m³ of seawater.

Mass of ship:

$$m = \rho V = 1025 \times 20000 = 20.5 \times 10^6 \text{ kg}$$

If 500,000 kg cargo is added:

$$\Delta V = \frac{\Delta m}{\rho} = \frac{500000}{1025} = 488 \text{ m}^3$$ $$\Delta h = \frac{\Delta V}{A} = \frac{488}{2000} = 0.244 \text{ m (sinks 24.4 cm deeper)}$$

Example 5: Apparent Weight

A 5 kg iron block ($\rho = 7870$ kg/m³) is submerged in water.

$$V = \frac{m}{\rho} = \frac{5}{7870} = 6.35 \times 10^{-4} \text{ m}^3$$ $$F_b = \rho_{\text{water}} \times V \times g = 1000 \times 6.35 \times 10^{-4} \times 9.8 = 6.22 \text{ N}$$ $$W_{\text{apparent}} = mg - F_b = 49 - 6.22 = 42.8 \text{ N}$$

Example 6: Hot Air Balloon

A balloon (volume = 500 m³) contains hot air at 100°C ($\rho = 0.95$ kg/m³). Outside air is at 20°C ($\rho = 1.20$ kg/m³). Find lift.

$$F_b = \rho_{\text{outside}} \times V \times g = 1.20 \times 500 \times 9.8 = 5880 \text{ N}$$ $$W_{\text{hot air}} = \rho_{\text{hot}} \times V \times g = 0.95 \times 500 \times 9.8 = 4655 \text{ N}$$ $$\text{Lift} = F_b - W_{\text{hot air}} = 1225 \text{ N}$$

This can lift a payload of 125 kg!

Applications

• Ships and submarines: Controlled buoyancy
• Hot air balloons: Less dense gas rises
• Fish swim bladders: Adjust density to hover
• Hydrometers: Measure fluid density
• Life jackets: Increase buoyancy to float