# Archimedes Principle

The Statement of Archimedes Principle

Any fluid applies equal pressure in every direction. This pressure is the result of the weight of the fluid. When an object is partially or completely submerged in a fluid, it exerts an upward force on the object. This upward force is called the buoyant force. Due to the buoyant force, there is an apparent decrease in the weight of the object. The decreased weight is equal to the weight of the fluid, displaced by the object. This relation was invented by Archimedes. From large ships to small boats, aircraft, submarines all of these operate according to the principle of buoyancy.

Apparent Weight

The original weight of an object acts downwards through its center of gravity. When an object is immersed in a fluid, an upward thrust namely buoyant force is exerted on the object. Due to this upward force, the resultant downward force decreases and the object feels lighter. If the object floats on the surface, it is effectively weightless. The apparent decrease in the weight is equal to the magnitude of the upward buoyant force.

The apparent weight of an object is given by the difference between the actual weight and the buoyant force.

Archimedes Principle Derivation

The principle is based on the buoyancy principle, which states that a gas or liquid can exert an upward force on any object, fully or partially immersed in it. The upward thrust is called the buoyant force.

In the diagram above, a cylinder of height h and radius r is immersed vertically in a liquid such that its flat surfaces are at depths h₁ and h₂ with h₁ <  h₂. The liquid exerts a perpendicular thrust (pressure) at each point on the surface of the cylinder. Due to the axial symmetry, the net thrust on the curved surface is zero. The downward pressure on the upper flat surface is h₁ρg, where ρ is the density of the liquid and g is the gravitational acceleration. The upward pressure on the lower flat surface is h₂ρg. If the atmospheric pressure is P$_{atm}$, the downward force on the upper surface,

F₁ = (P$_{atm}$ + h₁ρg)πr²

The upward force on the lower surface is,

F₂ =  (P$_{atm}$ + h₂ρg)πr²

Since, h₁ <  h₂ the resultant force on the cylinder is upward and the magnitude is,

F$_{b}$ = F₂ - F₁

F$_{b}$ = (h₂ - h₁)ρgπr²

Since h = h₂ - h₁ is the height of the cylinder and V = πr²h is its volume, the upward thrust can be expressed as,

F$_{b}$ = ρVg

The right-hand side of this upthrust formula is nothing but the weight of liquid of equal volume V as the submerged object. The magnitude of the buoyant force is, however, equal to the apparent decrease in the weight of the object. Therefore,

Apparent decrease of the object’s weight = weight of the fluid displaced by the object

Law of Floating

Whether an immersed object will float or sink, depends on the magnitudes of the actual weight W₁ of the object and the buoyant force W₂ exerted by the fluid.

• W₁ > W₂ :  The resultant force on the object is downwards, causing it to sink. When the density of the object is greater than that of the fluid, this condition arises.

• W₁ = W₂ : When the densities of the object and the fluid are equal, the actual weight and the buoyant force become equal. The object can float at any depth in a fully submerged state.

• W₁ < W₂ : The net force acts in the upward direction leading to a partially submerged condition of the object. The density of the object is less than the fluid in such cases.

Application of Archimedes Principle

• Using Archimedes law, the volume or density of any rigid body can be computed. The proportions of the constituent metals of an alloy can be easily calculated using this principle.

• Submarines operate using the Archimedes theory. It has a large ballast tank, which controls the depth of the marine. By adjusting the quantity of water in the ballast tank, the actual weight of the submarine is varied and thus the desired depth can be achieved.

• Ships are made hollow such that the effective density is less than the density of water. Due to the buoyant force having greater magnitude than the ship’s weight, the ship can float in a partially submerged state. Aircraft are made using the same concept.

• The densities of liquids are computed using hydrometers which work according to the Archimedes principle of buoyancy.

• Hot air balloons can float in the air because the density of hot air is less than the density of ambient cool air.

Did you know?

• Archimedes of Syracuse introduced the theory of buoyancy in his book On Floating Bodies (written in the Greek language) around 250 BC. This theory is considered to be the cornerstone in the study of hydrostatics.

• It is reported that Archimedes called out “Eureka”, meaning “I have found (it)” when he finally comprehended how to detect if a crown is made of impure gold using the theory of buoyancy.

• A floating body does not have any apparent weight.

• Surface tension or capillarity effect is not incorporated with the Archimedes principle.

• A large lunar impact crater is named after Archimedes.

• A portrait of Archimedes is engraved on the prestigious “Fields Medal”.