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 relationship was invented by Archimedes. From large ships to small boats, aircraft, submarines all of these operate according to the principle of buoyancy. Archimedes’s principle is extremely important in the study of buoyancy in physics, this principle is explained very briefly in Chapter 10 called mechanical properties of fluids of Class 11 in the NCERT book, this chapter is prescribed by the Central Board of secondary education as it includes various important topics whose understanding is extremely important for higher studies.
The chapter mechanical properties of fluids mainly deal with the common physical properties of liquids and gases.
The study of fluid is extremely important as fluids cover a significant part of the earth and they are used in our day to day lives as an essential component. Every mammal’s body on the earth constitutes mostly water. All the processes that are responsible for the growth of living beings are mediated by fluids.
The topics related to Archimedes principle have taught us that volume of solids can be changed by stress. The volume of all the three forms that Is solid liquid or gas is dependent on the stress or pressure acting upon it. Fixed volume in physics means the volume under atmospheric pressure. Solids and liquids as compared to gases have much lower compressibility.
Fluids offer very little resistance to shear stress. The shear stress of fluids is about a million times smaller than the shear stress of solids.
Archimedes’s principle is also known as the physical law of buoyancy; it was discovered by the Asian Greek mathematician Archimedes who was a Greek philosopher, scientist and engineer. Eureka is a word popularised by Archimedes. He exclaimed Eureka when he realised he had invented the method of detecting if something is made of pure or impure gold. In the widespread tale, Archimedes didn’t use his principle he only used displaced water to measure the volume of the crown, an alternative approach is applied with the use of this principle - A scale has to be balanced after placing a crown on one side and pure gold on the other, submerge the scale in water, According to Archimedes principle, if the crown’s density differs from pure gold’s then the scale will get out of balance underwater.
Apparent Weight
The original weight of an object acts downwards through its centre 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.
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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.
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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 isP\[_{atm}\], the downward force on the upper surface,
F\[_{1}\] = \[\left ( P_{atm}+h_{1}\rho g \right )\]\[\pi\] r\[^{2}\]
The upward force on the lower surface is,
F\[_{2}\] = \[\left ( P_{atm}+h_{2}\rho g \right )\]\[\pi\] r\[^{2}\]
Since, h₁ < h₂ the resultant force on the cylinder is upward and the magnitude is,
F\[_{b}\] = F\[_{2}\]-F\[_{1}\]
F\[_{b}\] = \[\left ( h_{2}-h_{1} \right )\]\[\rho\] g \[\pi\] r\[^{2}\]
Since h = h₂ - h₁ is the height of the cylinder and V = πr²h is its volume, the upward thrust can be expressed as,
F\[_{b}\] = \[\rho\] Vg
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.
Key Concepts related to Archimedes Principle are as Follows-
Pressure
Streamline flow
Bernoulli’s principle
Reynolds number
Surface tension
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”.
FAQs on Archimedes Principle
1. State Archimedes Principle.
Archimedes’s principle states that – ‘the loss of weight of a body submerged (partially or fully) in a fluid is equal to the weight of the fluid displaced’.
Mathematically written as:
Fb = ρ x g x V
2. What are the applications of Archimedes’ principle?
The related density of a substance can be determined by using Archimedes principle.
The process of designing ships and submarines requires the use of Archimedes' principle.
The hydrometres work according to the Archimedes principle of buoyancy which is used to compute the densities of liquids.
Beach Ball float on the surface of the water because of the Archimedes principle of buoyancy. While pushing the beach ball into the water, the buoyant force increases which in turn doesn’t let the beach ball sink.
The working of the submarines also depends on the Archemedes’s Principle. Submarine scan lot and submerged into the water by maintaining the density of the water displeased and submarine.
3. Where can I find the detailed analysis of Archimedes principle?
A detailed analysis of Archimedes’ principle can be studied by visiting Vedantu’s website which gives a comprehensive understanding of the topic as it is written in an extremely simplified language which enables students to get a good grade in the examination.
4. What is the law of floating?
According to the law of floating- the magnitude of the actual weight W1 of a particular object and the buoyant force W2 which is exerted by the fluid is the reason why an object submerged in water, floats or sinks.
W₁ > W₂: The resultant force on the object is downwards, causing it to sink. This happens when the density is greater than the density of the fluid.
An object is capable of floating at any depth only when the density and fluid of the object are equal, this is also when the actual weight and buoyant force also become equal. It can be represented as- W₁ = W₂
W₁ < W₂: The net force acts in the upward direction leading to a partially submerged condition of the object.in such a situation the density of the object is less than the density of the fluid.
5. What is the derivation of Archimedes’s principle?
The mass of the liquid displaced is -
Mass = Density × volume
= ρ × V
This can be explained by explaining the definition of density-
Density, ρ = mass/volume = m/v
Weight of the displaced liquid-
Weight = mass x acceleration due to gravity
W = M x G = ρ × V × g
From Archimedes’s principle, we get-
The apparent loss of weight = weight of water displaced = ρ × V × g
Thus, the Thrust force is,
Thrust = ρ × V × g
Where,
ρ = density of the liquid,
V = volume of liquid displaced.
6. Who is Archimedes?
Archimedes was a Greek mathematician, physicist, astronomer, engineer, and inventor, born in Syracuse of Sicily. He gave the theory of buoyancy and started the study of hydrostatics. He proved various geometrical theorems (e.g. volume and area of a sphere, area of a circle) using the concept of “infinitesimals” far before the advent of modern calculus. One notable invention of his was the Archimedes’ Screw, which was used to pump water from low lying areas to irrigation channels.