Buoyancy Formula: Derivation, Relative Density & Dimension

Buoyancy is the force acting upwards on the body immersed in a fluid partially or completely. This is also called upthrust, buoyant force, or simply buoyancy. 

• A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces.
• Its SI Unit is Newton as it is a kind of force.
• The line of action of the buoyant force passes through the centre of the volume of the displaced body.
• The Buoyancy force is equal to "hρga" where h is the height up to which the body is immersed, is the density of the object, g is the acceleration due to gravity, and ‘a’ is the area of cross-section of the object.

Read Also: Venturimeter

Key Terms: Buoyancy, Buoyant Force, Upthrust, Upward force, Buoyancy formula, Relative Density, Archimedes Principle

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What is Buoyancy?

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A body, when immersed in a fluid, partially or wholly displaces the fluid due to which the fluid tries to regain its original position and exerts an upward force on the body known as Buoyant force or simply Buoyancy.

The pressure exerted on the bottom of the object in a fluid is greater than the pressure at the top of the object leading to a pressure difference. The pressure difference between the top and the bottom of the object results in a net upward force or buoyant force.

• The process of acting buoyant force on the object is called Buoyancy.
• It is the upward force acting on the body immersed in the fluid. It is also called upthrust because the force acts in an upward direction.
• The buoyant force acts to prevent the object from sinking.
• The buoyant force acts at the centre of the object.

Buoyancy Meaning

• For a solid body of uniform density, the centre of gravity coincides with the centre of buoyancy.

• The SI unit of Buoyant Force is ‘Newton’ expressed by ‘N’.
• The CGS unit for Buoyant Force is ‘dyne’.
• The dimensional formula for Buoyant force is the same as of Force [MLT ^-2].

Check More: Density Vs Specific Gravity

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Buoyancy Formula

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This Buoyant Force acting on the object immersed in the fluid is equal to the product of the Pressure difference between the top and the bottom of the object immersed in the fluid and the Base Area of the object.

The pressure difference depends on the vertical distance ‘h’ between the top and bottom of the object, mass density of the fluid 'ρ’, and acceleration due to gravity ‘g’.

Buoyant Force = Pressure difference × Area of base of the body

Thus,

Buoyant Force = hρgA

Read more: Mechanical Properties of Fluids

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Buoyancy Formula Derivation

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Consider a fluid of density ‘ρ' at rest in a container. Let's assume a cylindrical element of fluid having an area of base ‘A’ and height ‘h’. The resultant horizontal force gets eliminated and the resultant vertical forces should balance the weight of the element.

The force acting in the vertical direction is due to the difference in the force acting on the top and the bottom of the cylindrical element.

Hence,

Net Force on the Element=Weight of the Cylindrical Element

Pressure Difference × Base Area=Weight of the Cylindrical Element

→ (P₂ – P₁) A = mg

→( P₂ – P₁) A = (ρv)g

→(P₂ – P₁)A = ρAhg

→ (P₂ – P₁) = hρg

Buoyant Force=Pressure difference × Base Area

Buoyant Force = hρgA

Where, m = mass of the element

Read More: Fluid Mechanics Formula

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Archimedes' Principle of Buoyancy

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Archimedes' Principle of Buoyancy reveals the idea of the apparent weight of the body immersed in the liquid. The principle is described as “when a body is immersed partly or wholly in a liquid at rest, it losses some of its weight and is buoyed up by a force equal to the weight of the liquid displaced by the immersed part of the body”.

Apparent Immersed Weight =Weight – Weight of Displaced Liquid

Apparent Immersed Weight (W') = W (1 – \(\frac{ρ }{σ}\))

Or, 

Apparent Immersed Weight W' = W (1- \(\frac{1}{Relative Density}\) )

Where, 'ρ' is the density of liquid, 'σ' is the density of the object immersed, and ‘W’ is the True Weight of the Object.

The video below explains this:

Archimedes Principle Detailed Video Explanation:

Read More:

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Applications of Archimedes Principle

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Archimedes' principle is applied in many day-to-day activities. Some of the best examples has been curated below.

Helium balloon

A helium balloon in a moving car experiences buoyant force by the air mass moving in the opposite direction to the car’s acceleration. For the same reason, as the car goes around a curve, the balloon will drift toward the inside of the curve.

Submarine and Ships

Submarines and ships are designed so that it can displace the fluid equal to the weight of the ship or submarine.

Fish

A fish can regulate its density by expanding or contracting an inbuilt air sac that changes its volume. The fish can move upward by increasing its volume leading to a decrease in its density and can move downward by contracting its volume leading to an increase in its density. Hence, The fish moves up and down in the water with the variation of its relative density.

Air Vessels

A dirigible, if displaces more amount of air, it rises; if it displaces less, it falls; and If it displaces exactly its weight, it hovers at a constant altitude.

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Relative Density

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Relative Density is the ratio of the density of a substance to the ratio of the density of another reference material. Relative density is also the specific gravity when measured with respect to water at its densest which is 40 C.

Relative Density= \(\frac{ Density of the Substance }{Density of the reference material}\)

Or, 

Relative Density= \(\frac{ Density of the Substance }{Density of water at 4℃}\)

Read More: Hydrostatic Paradox

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Law of Floatation

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The three Law of floatation explains why a body when immersed in the liquid tends to flow or tends to sink. 

The Body will Sink

In the case when the average density of the object is greater than that of the fluid, its relative density becomes greater than 1 and the object tends to sink.

If, σ > ρ, \(\frac{σ}{ρ }\) >1, The body will sink

The Body will Float

In the case when the average density of the object is less than that of the fluid, its relative density becomes less than 1 and the object tends to float.

If, σ < ρ, \(\frac{σ}{ρ }\) <1, The body will Float

The Body will Float in Equilibrium State

In the case when the average density of the object is equal to that of the fluid, its relative density becomes equal to 1 and the object tends to will keep floating fully submerged in an equilibrium state. 

If, σ= ρ, \(\frac{σ}{ρ }\) =1, The body will Float in equilibrium state.

Check Out:

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Things to Remember

• The buoyant force acts at the centre of the object.
• The principle of floatation says that a body displaces a weight of liquid equal to its weight.
• Archimedes' principle can be expressed in terms of volume i.e. ρgV
• Buoyancy force = weight of an object in space − the weight of an object immersed in the fluid.
• Relative Density, also known as specific gravity, plays an important role in floating a body.
• The density of the fluid, the volume of the object, and acceleration due to gravity are the factors that affect the buoyant force.

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Previous Year Questions

1. The cylindrical tube of a spray pump has radius… (NEET 2015)
2. Water rises to a height h in capillary tube… (NEET 2015)
3. A ball whose density is 0.4×103kg/m3 falls into water …. (BITSAT 2007)
4. A small sphere of radius ′r′ falls from rest in a viscous liquid...(NEET 2018)
5. A rectangular film of liquid is extended from...(NEET 2016)
6. A boat with scrap iron in it is floating in a small tank of water… (JIPMER 1998)
7. A certain number of spherical drops of a liquid of radius… (NEET 2014)
8. A fluid is in streamline flow across a horizontal pipe of….(NEET 2013)