Buoyancy, Density & Archimedes Principle

Buoyancy: Core Idea

• Buoyancy is the upward (vertical) force exerted by a fluid on an object immersed (fully or partially) in it.

• It opposes the object’s weight; whenever an object is placed in a fluid two main forces act:
  • Weight ( $W = mg$ , downward).
  • Buoyant force ( $F_b$ , upward).

• Source of $F_b$: pressure at the lower surface is greater than pressure at the upper surface; the net result is an upward force.

• Formal equation (Archimedes):
  $F<em>b = \rho</em>{fluid} \; V<em>{displaced} \; g$ where • $\rho</em>{fluid}$ = density of the fluid (kg/m³),
  • $V_{displaced}$ = volume of fluid displaced by the object (m³),
  • $g$ = acceleration due to gravity (≈ $9.81\;\text{m/s}^2$).

Qualitative Outcomes

• F_b > W → Positive buoyancy → object rises / floats.

• $F_b = W$ → Neutral buoyancy → object stays suspended at current depth ("flinks").

• F_b < W → Negative buoyancy → object sinks.

Everyday Example

• A solid steel block sinks, but a steel ship (same mass) floats because the ship’s hull traps air, enlarging $V<em>{displaced}$; this increases $F</em>b$ until it equals the ship’s weight.

Density ( $\rho$ )

• Definition: mass per unit volume.
  $\rho = \dfrac{m}{V}$

• SI unit: kg/m³.

• Common symbol: Greek letter $\rho$ (rho).

Typical Densities

• Air: $1.3\;\text{kg/m}^3\;(0.0013\;\text{g/cm}^3)$

• Expanded polystyrene: $14\;\text{kg/m}^3\;(0.014\;\text{g/cm}^3)$

• Wood (beech): $750\;\text{kg/m}^3\;(0.75\;\text{g/cm}^3)$

• Petrol: $800\;\text{kg/m}^3\;(0.80\;\text{g/cm}^3)$

• Ice (0 °C): $920\;\text{kg/m}^3\;(0.92\;\text{g/cm}^3)$

• Polythene: $950\;\text{kg/m}^3\;(0.95\;\text{g/cm}^3)$

• Water (4 °C): $1000\;\text{kg/m}^3\;(1.0\;\text{g/cm}^3)$

• Granite: $2700\;\text{kg/m}^3\;(2.7\;\text{g/cm}^3)$

• Aluminium: $2700\;\text{kg/m}^3\;(2.7\;\text{g/cm}^3)$

• Stainless steel: $7800\;\text{kg/m}^3\;(7.8\;\text{g/cm}^3)$

• Copper: $8900\;\text{kg/m}^3\;(8.9\;\text{g/cm}^3)$

• Lead: $11400\;\text{kg/m}^3\;(11.4\;\text{g/cm}^3)$

• Mercury: $13600\;\text{kg/m}^3\;(13.6\;\text{g/cm}^3)$

• Gold: $19300\;\text{kg/m}^3\;(19.3\;\text{g/cm}^3)$

• Concrete: $2400\;\text{kg/m}^3\;(2.4\;\text{g/cm}^3)$

• Glass (average): $2500\;\text{kg/m}^3\;(2.5\;\text{g/cm}^3)$

• Platinum: $21500\;\text{kg/m}^3\;(21.5\;\text{g/cm}^3)$

• Osmium: $22600\;\text{kg/m}^3\;(22.6\;\text{g/cm}^3)$

Density & Floating Rules

• If \rho{object} < \rho{fluid} → object floats.

• If \rho{object} > \rho{fluid} → object sinks.

• If $\rho<em>{object} = \rho</em>{fluid}$ → object "flinks" (neutral).

Specific Gravity / Relative Density

• Definition: ratio of the density of a substance to the density of a standard substance (usually water at 4 °C for liquids/solids, air for gases).
  $\text{SG} = \dfrac{\rho<em>{substance}}{\rho</em>{water}}$

• Dimensionless (no units).

Worked Example

• A 200 g sample occupies 50 cm³.
  • Density: $\rho = \dfrac{0.200\;\text{kg}}{50\times10^{-6}\;\text{m}^3} = 4000\;\text{kg/m}^3$
  • Specific gravity: $\text{SG} = \dfrac{4000}{1000} = 4$.

Volume

• Defined as the amount of space occupied by an object.

• SI unit: m³; common lab unit: cm³ or mL.

Water-Displacement Method

• Measure initial water level $V<em>1$, submerge object, read new level $V</em>2$.

• Object’s volume: $V<em>{object} = V</em>2 - V_1$.

• Archimedes used this to test the purity of a gold crown.

Archimedes’ Principle

• "A body wholly or partly submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced."
  $F<em>b = W</em>{displaced} = m<em>{displaced} g = \rho</em>{fluid} V_{displaced} g$.

Types of Buoyancy (Visualized)

• Positive: F_b > W (e.g., life jacket).

• Neutral: $F_b = W$ (e.g., fish controlling swim bladder, scuba diver at hover).

• Negative: F_b < W (e.g., stone in water).

Factors Affecting Buoyancy

• Density of the object (higher → more likely to sink).

• Density of the fluid (salt water > fresh water, so objects float easier in oceans or the Dead Sea).

• Volume displaced (larger/flatter shapes displace more water, improving buoyancy — reason ships are wide).

• Shape (hull design, surface area distribution).

• Gravity (g) (lower g → easier to float; relevant for space-habitation aquatics).

• Temperature of the fluid (warmer → lower $\rho_{fluid}$ → reduced buoyancy).

Sample Problems

1. Lifebuoy Check
   Data: $\rho<em>{buoy}=200\;\text{kg/m}^3$, $V=0.15\;\text{m}^3$, $\rho</em>{seawater}=1025\;\text{kg/m}^3$.
   • Weight of buoy: $W = \rho<em>{buoy} V g = 200\times0.15\times9.81 = 294\;\text{N}$. • Max buoyant force: $F</em>b= \rho<em>{seawater} V g = 1025\times0.15\times9.81 \approx 1506\;\text{N}$. • Fb > W → lifebuoy will float with large margin of safety.

2. Submerged Wooden Block
   Data: $V = 0.05\;\text{m}^3$, $\rho<em>{water}=1000\;\text{kg/m}^3$. $F</em>b = \rho V g = 1000 \times 0.05 \times 9.81 = 490.5\;\text{N}$.
   Interpretation: this is the upward force resisting the block’s weight.

3. Which 90 kg Person Floats?
   Two people both weigh 90 kg. Person A spreads out (star-fish), person B curls tightly. Both displace enough water to equal 90 kg, but A achieves larger surface area → larger $V_{displaced}$ before submerging (lung capacity + shape). Thus A is more likely to float/neutrally hover.

Practical / Ethical / Real-World Links

• Naval architecture: hull design must ensure positive buoyancy even when taking on water.

• Safety equipment: life vests & lifebuoys must have density well below 1000 kg/m³; regulations demand a safety factor.

• Environmental concern: Dense pollutants (e.g., mercury) sink, affecting benthic ecosystems; lighter oils float, impacting surface wildlife.

• Medical application: Hydrometers use specific gravity to test urine (diagnostics) and battery acid (maintenance).

• Philosophy of measurement: Archimedes pioneered empirical testing over authority (gold crown story).

Summary Equations Cheat-Sheet

• Density: $\rho = \dfrac{m}{V}$.

• Buoyant force: $F<em>b = \rho</em>{fluid} V_{displaced} g$.

• Archimedes principle (alternate): $F<em>b = W</em>{displaced}$.

• Specific gravity: $\text{SG} = \dfrac{\rho<em>{substance}}{\rho</em>{water}}$ (dimensionless).

• Weight: $W = mg$.

Quick Study Tips

• Always identify what is displacing the fluid (object + air pockets).

• Compare densities first; only if they’re close, analyze shapes & temperature.

• Remember: 1 mL = 1 cm³ = $1\times10^{-6}\,\text{m}^3$; useful in lab computations.

• Practice with mixed units; convert everything to SI before plugging into formulas.