Relative Density and Specific Gravity

Relative density (RD) aur specific gravity (SG) fluid mechanics aur buoyancy ke fundamental concepts hain, jo objects immersed in fluids ke behavior ko quantify karte hain. Ye concepts ship design, floating bodies, liquid density measurement, and fluid engineering me extensively use hote hain.

1. Introduction

Buoyancy:

Upward force exerted by a fluid on an object immersed in it, opposing the weight of the object.

Archimedes’ Principle:

A body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by it.

Relative Density (RD) / Specific Gravity (SG) provide dimensionless measures of how heavy a substance is compared to a reference fluid, usually water.

• Determines whether an object floats or sinks
• Determines submerged fraction of floating body
• Basis for hydrometer readings, fluid separation, and density-based analysis

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2. Definition of Relative Density and Specific Gravity

2.1 Relative Density (RD)

Relative density of a substance is the ratio of its density to the density of a reference substance (water at 4°C for liquids).

RD=ρsubstanceρreferenceRD = \frac{\rho_{substance}}{\rho_{reference}}RD=ρreference​ρsubstance​​

• Dimensionless quantity
• RD > 1 → heavier than reference
• RD < 1 → lighter than reference

2.2 Specific Gravity (SG)

• Specific gravity is often used interchangeably with RD

SG=weight of substanceweight of equal volume of waterSG = \frac{\text{weight of substance}}{\text{weight of equal volume of water}}SG=weight of equal volume of waterweight of substance​

• SG = RD for solids and liquids under standard conditions
• Important for buoyancy analysis and fluid selection

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3. Relation with Buoyancy

• Buoyant force:

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

• For floating body at equilibrium:

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

• Fraction submerged:

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

• SG / RD directly determines floatation and immersed volume fraction

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4. Mathematical Derivation

4.1 Fully Submerged Body

• Body volume: VbodyV_{body}Vbody​
• Body density: ρbody\rho_{body}ρbody​
• Fluid density: ρfluid\rho_{fluid}ρfluid​

Weight of body: W=ρbodygVbodyW = \rho_{body} g V_{body}W=ρbody​gVbody​

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

• Floating condition: Fb≥WF_b \ge WFb​≥W → ρfluid≥ρbody\rho_{fluid} \ge \rho_{body}ρfluid​≥ρbody​
• Sinking condition: Fb<WF_b < WFb​<W → ρfluid<ρbody\rho_{fluid} < \rho_{body}ρfluid​<ρbody​

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4.2 Partially Submerged Body (Floating)

• Submerged volume: VsubV_{sub}Vsub​
• Equilibrium: Fb=WbodyF_b = W_{body}Fb​=Wbody​

ρfluidgVsub=ρbodygVbody  ⟹  VsubVbody=ρbodyρfluid=RD\rho_{fluid} g V_{sub} = \rho_{body} g V_{body} \implies \frac{V_{sub}}{V_{body}} = \frac{\rho_{body}}{\rho_{fluid}} = RDρfluid​gVsub​=ρbody​gVbody​⟹Vbody​Vsub​​=ρfluid​ρbody​​=RD

• Directly relates RD/SG to submerged fraction

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5. Practical Applications

5.1 Ships and Boats

• Determine draft and displacement
• Fraction submerged = RD of ship material / seawater
• Helps in load capacity calculation

5.2 Hydrometers

• Device to measure liquid density
• Floats higher in less dense liquids, lower in denser liquids
• Scale marked in SG or RD

5.3 Life Jackets and Buoyancy Aids

• Use materials with SG < 1
• Ensure sufficient buoyant force to float human body

5.4 Hot Air Balloons

• Gas density vs air density determines buoyant lift
• RD < 1 → balloon rises

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6. Examples and Calculations

Example 1: Wooden Block in Water

• Wood density: 600 kg/m³, water density: 1000 kg/m³
• Relative density:

RD=6001000=0.6RD = \frac{600}{1000} = 0.6RD=1000600​=0.6

• Fraction submerged = 0.6 → 60% submerged

Example 2: Metal Cube in Water

• Cube density: 7800 kg/m³
• Water density: 1000 kg/m³
• RD = 7800 / 1000 = 7.8 → cube sinks completely

Example 3: Floating Cylinder

• Cylinder volume: 2 m³, density: 800 kg/m³
• Fluid: water (1000 kg/m³)
• Submerged volume:

Vsub=Vbody⋅RD=2⋅0.8=1.6m3V_{sub} = V_{body} \cdot RD = 2 \cdot 0.8 = 1.6 m^3Vsub​=Vbody​⋅RD=2⋅0.8=1.6m3

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7. Effect of Fluid Density

• Denser fluids → less submerged fraction
• Example: Wood in freshwater vs seawater (ρ ≈ 1025 kg/m³)
• Submerged fraction decreases → floats higher
• Equation:

Vsub=VbodyρbodyρfluidV_{sub} = V_{body} \frac{\rho_{body}}{\rho_{fluid}}Vsub​=Vbody​ρfluid​ρbody​​

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8. Buoyancy in Liquids of Different Densities

• Multiple fluid layers → compute cumulative buoyant force

Fb=g∑(ρiVi)F_b = g \sum (\rho_i V_i)Fb​=g∑(ρi​Vi​)

• Used in stratified tanks and oil-water separation

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9. Stability Considerations

• Stability depends on center of gravity (G), center of buoyancy (B), and metacenter (M)
• GM = M – G → stable if GM > 0
• RD affects draft, center of buoyancy, and floating orientation
• Important for ships, buoys, and floating platforms

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10. Graphical Representation

1. Submerged Fraction vs RD → linear graph
2. Buoyant Force vs Submerged Volume → linear relationship
3. Stability diagrams → metacentric height and RD effect

V_sub/V_body
|
|        *
|       *
|      *
|     *
|_____*________ RD

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11. Measurement Techniques

11.1 Hydrometer

• Measures SG of liquid
• Principle: Archimedes’ principle → floating height varies with density

11.2 Pycnometer

• Known volume container
• Measure mass of liquid → density → SG

SG=ρliquidρwaterSG = \frac{\rho_{liquid}}{\rho_{water}}SG=ρwater​ρliquid​​

11.3 Weighing Method

• Weigh object in air and fluid
• Buoyant force = weight difference

SG=WairWair−WfluidSG = \frac{W_{air}}{W_{air} – W_{fluid}}SG=Wair​−Wfluid​Wair​​

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12. Relative Density in Real-Life Applications

1. Petroleum Industry: Classify oils and fuels using SG
2. Food Industry: Sugar solutions, syrups, density measurement
3. Civil Engineering: Soil density vs water → buoyancy of submerged structures
4. Marine Engineering: Ship design, load capacity, stability
5. Mining: Mineral separation → denser minerals sink, lighter float

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13. Numerical Problems

Problem 1: Submerged Fraction

• Wood cube, ρ = 700 kg/m³, water ρ = 1000 kg/m³
• Fraction submerged:

Vsub/Vcube=700/1000=0.7 (70%)V_{sub}/V_{cube} = 700/1000 = 0.7 \, (70\%)Vsub​/Vcube​=700/1000=0.7(70%)

Problem 2: Buoyant Force on Sphere

• Sphere volume: 0.5 m³, water density: 1000 kg/m³

Fb=ρgV=1000⋅9.81⋅0.5=4905NF_b = \rho g V = 1000 \cdot 9.81 \cdot 0.5 = 4905 NFb​=ρgV=1000⋅9.81⋅0.5=4905N

• If sphere weight = 4000 N → floats, 0.4 m³ submerged

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14. Summary Table

Quantity	Formula / Relation	Notes
Buoyant Force (F_b)	F_b = ρ_fluid g V_displaced	Acts upward
Relative Density (RD)	RD = ρ_body / ρ_fluid	Dimensionless
Specific Gravity (SG)	SG = Weight_body / Weight_equal_volume_water	Often equal to RD
Submerged fraction	V_sub / V_body = RD	Determines floating height
Stability (Metacentric height)	GM = M – G	GM > 0 → stable