Buoyancy and Archimedes’ Principle

Buoyancy aur Archimedes’ Principle fluid mechanics ke fundamental concepts hain, jo objects immersed in fluids ke behavior ko describe karte hain. Ye concepts ship design, submarine operation, hydraulic engineering, aerostatics, and fluid system analysis me extensively use hote hain.

1. Introduction

Fluid mechanics me, fluids (liquids aur gases) ka behavior study kiya jata hai. Jab object fluid me immerse hota hai, toh uspar upward force act karta hai, jo uske weight ko counter karta hai. Ye force ko buoyant force kehte hain.

Archimedes’ Principle is phenomenon ko describe karta hai mathematically:

“A body immersed in a fluid experiences a buoyant force equal to the weight of the fluid displaced by the body.”

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2. Basic Concepts

2.1 Buoyant Force

• Definition: Upward force exerted by fluid on submerged object
• Symbol: FbF_bFb​
• Unit: Newton (N)

Formula: Fb=ρfluidgVdisplacedF_b = \rho_{fluid} g V_{displaced}Fb​=ρfluid​gVdisplaced​

Where:

• ρfluid\rho_{fluid}ρfluid​ = density of fluid
• VdisplacedV_{displaced}Vdisplaced​ = volume of fluid displaced by body
• g = acceleration due to gravity

Direction: Always opposite to gravity, acts vertically upward through the center of buoyancy.

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2.2 Center of Buoyancy

• Point through which buoyant force acts
• Coincides with centroid of displaced volume
• Important for stability analysis of floating bodies

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3. Archimedes’ Principle

Statement:

“Any body wholly or partially immersed in a fluid experiences an upthrust equal to the weight of the fluid displaced by it.”

3.1 Mathematical Expression

For a body of volume VVV submerged in a fluid of density ρ\rhoρ: Fb=ρgVF_b = \rho g VFb​=ρgV

• For floating bodies, Fb=Wbody=ρbodygVbodyF_b = W_{body} = \rho_{body} g V_{body}Fb​=Wbody​=ρbody​gVbody​
• Determines whether object floats or sinks

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4. Derivation of Buoyant Force

4.1 For Submerged Object

• Consider a cubical element of fluid: height h, cross-sectional area A
• Pressure at top: P1=ρgh1P_1 = \rho g h_1P1​=ρgh1​
• Pressure at bottom: P2=ρgh2P_2 = \rho g h_2P2​=ρgh2​

Net upward force: Fb=P2A−P1A=ρg(h2−h1)A=ρgVF_b = P_2 A – P_1 A = \rho g (h_2 – h_1) A = \rho g VFb​=P2​A−P1​A=ρg(h2​−h1​)A=ρgV

• V = volume of fluid displaced
• Direction → upward

Conclusion: Buoyant force arises due to pressure difference between top and bottom surfaces.

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4.2 General Shape

• For irregular body, divide volume into small elements
• Sum of upward forces → total buoyant force = weight of fluid displaced
• Acts through center of buoyancy

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5. Floating and Submerged Bodies

5.1 Fully Submerged

• Object completely under fluid
• Buoyant force:

Fb=ρfluidgVbodyF_b = \rho_{fluid} g V_{body}Fb​=ρfluid​gVbody​

• Condition for sinking: Fb<WbodyF_b < W_{body}Fb​<Wbody​
• Condition for floating: Fb>WbodyF_b > W_{body}Fb​>Wbody​

5.2 Partially Submerged (Floating)

• Object floats at equilibrium:

Fb=Wbody=ρbodygVsubmergedF_b = W_{body} = \rho_{body} g V_{submerged}Fb​=Wbody​=ρbody​gVsubmerged​

• Fraction submerged:

VsubmergedVbody=ρbodyρfluid\frac{V_{submerged}}{V_{body}} = \frac{\rho_{body}}{\rho_{fluid}}Vbody​Vsubmerged​​=ρfluid​ρbody​​

• Determines floating height of ships, boats, and logs

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6. Relative Density (Specific Gravity)

• Specific Gravity:

SG=ρbodyρfluidSG = \frac{\rho_{body}}{\rho_{fluid}}SG=ρfluid​ρbody​​

• Floating condition: SG < 1
• Sinking condition: SG > 1
• Determines submerged fraction of object:

Submerged fraction=SG\text{Submerged fraction} = SGSubmerged fraction=SG

Example: Wood in water → SG = 0.6 → 60% submerged

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7. Stability of Floating Bodies

• Stability depends on metacenter
• Metacentric height (GM): distance between center of gravity (G) and metacenter (M)
• Stable equilibrium: M above G → body returns to original position if tilted
• Unstable equilibrium: M below G → body overturns
• Neutral equilibrium: M coincides with G → body remains tilted

Applications: Ship design, buoy design, floating platforms

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8. Applications of Buoyancy

1. Ship and Submarine Design
   • Determine draft, stability, load capacity
2. Hot Air Balloons and Airships
   • Gas density < air density → buoyant force lifts object
3. Hydrometers
   • Measure density of liquids using floating principle
4. Life Jackets
   • Provide enough buoyant force to float a human
5. Dams and Reservoirs
   • Buoyant force considered in design of submerged structures

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9. Solved Examples

Example 1: Floating Cube

• Cube volume V=0.1m3V = 0.1 m^3V=0.1m3, density 600kg/m3600 kg/m^3600kg/m3, water density 1000kg/m31000 kg/m^31000kg/m3
• Submerged volume fraction:

Vsub/V=ρcube/ρwater=600/1000=0.6V_{sub} / V = \rho_{cube}/\rho_{water} = 600/1000 = 0.6Vsub​/V=ρcube​/ρwater​=600/1000=0.6

• 60% of cube submerged, 40% above water

Example 2: Fully Submerged Sphere

• Sphere radius r = 0.5 m, density = 800 kg/m³, water density = 1000 kg/m³
• Volume:

V=43πr3=43π(0.5)3≈0.5236m3V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (0.5)^3 \approx 0.5236 m^3V=34​πr3=34​π(0.5)3≈0.5236m3

• Buoyant force:

Fb=ρgV=1000⋅9.81⋅0.5236≈5139NF_b = \rho g V = 1000 \cdot 9.81 \cdot 0.5236 \approx 5139 NFb​=ρgV=1000⋅9.81⋅0.5236≈5139N

• Weight of sphere:

W=800⋅9.81⋅0.5236≈4111NW = 800 \cdot 9.81 \cdot 0.5236 \approx 4111 NW=800⋅9.81⋅0.5236≈4111N

• Since F_b > W → sphere floats

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10. Effect of Density Variation

• Denser fluids → higher buoyant force for same object
• Mercury (ρ = 13600 kg/m³) → objects float more easily than in water
• Applications: Mineral separation, hydrometallurgy

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11. Gas Buoyancy

• Hot air balloon: buoyant force = weight of displaced air:

Fb=ρairgVballoonF_b = \rho_{air} g V_{balloon}Fb​=ρair​gVballoon​

• Condition to lift payload:

Fb>Wpayload+WballoonF_b > W_{payload} + W_{balloon}Fb​>Wpayload​+Wballoon​

• Temperature, air density, and gas type affect lift

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12. Graphical Representation

1. Buoyant Force vs Submerged Volume → linear relationship
2. Floating Fraction vs SG → linear, slope = 1
3. Metacentric height effect → stability diagrams

F_b
|
|        *
|       *
|      *
|     *
|____*________ V_sub

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13. Summary Table

Concept	Formula / Relation	Notes
Buoyant Force	F_b = ρ g V_displaced	Acts upward, through center of buoyancy
Floating condition	F_b = W_body	Submerged fraction = ρ_body / ρ_fluid
Fully submerged	F_b = ρ_fluid g V_body	Sinking if F_b < W_body
Relative density (SG)	SG = ρ_body / ρ_fluid	Determines floatation
Stability (Metacentric Height)	GM = M – G	GM > 0 → stable, GM < 0 → unstable