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Archimedes Principle

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The principle of objects submerged in fluids was discovered by Greek mathematician Archimedes. It depends on force of buoyancy. This force is not a mysterious force that exists in fluids but exists due to the pressure difference. Let study more about the buoyant force and how this force is responsible for the principle.  

Archimedes Principle of Buoyancy

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Archimedes Principle states that :

Any object completely or partially submerged in a fluid experiences upward thrust with the magnitude equal to the weight of the fluid displaced by the immersed part of object. This upward thrust is the force of buoyancy.

It is given as
Buoyant force acting on an object = Weight of the fluid displaced by the object.
If the density of the object is greater than the fluid the object will sink.
If the density of the object is equal to that of the fluid the object will sink or float.
If the density of the object is less than the fluid the object will float.

Archimedes Principle Formula

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Lets take a body immersed in the fluid as shown in figure,

Buoyancy Force

The buoyant force in fluids is a physical cause due to the pressure difference between the lower and upper sides of the object that is equal to the weight of the displaced fluid. If P2A acts upwards on the bottom of the surface and P1A and gravitational force due to object Mg acts on fluid in down ward direction then the buoyancy force B is given by,
B = P2A - P1A = Mg
Hence, the pressure difference is equal to weight of displaced fluid.The pressure in fluids increases with depth and acts equally in all direction.

The density in terms of buoyancy is defined as,
B = $\rho$Vg
where,
$\rho$ is density of fluid
V is volume of fluid
g is the gravitational force.

Archimedes Principle Experiment

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Archimedes Principle Experiment
Take a piece of metal tie it to a thread and attach it to spring balance. Note down the pointer reading. Now keeping the same setup, immerse the metal piece completely in water and note down the pointer reading. You could see the reading will be less when it is immersed compared to before that tells that the body (metal piece) is experiencing the upward thrust. Subtract the first reading from the second (immersed in fluid) given as
Wdifference = Wwater - Wair

where,
Wdifference is the weight of displaced fluid
Wfluid is the weight of the metal piece immersed in fluid
Wair is weight of the metal piece in air.

Example of Archimedes Principle

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Lets go through some examples of Archimedes principle:

How steel ship floats?
You often see a steel ship floats in water even though piece of steel sinks in it. To know what for it happens so let us apply Archimedes principle since the mass of ship is distributed over larger area of water we could see that the upthrust of water will be greater than mass of ship which makes the greater mass of ship to get displaced. So it floats.

Lighter toys float in water
Its often heard about the story of Archimedes who realized this principle when he was taking bath and experienced the upward thrust. Same way even we put some rubber toys in a tub water which have less density than the fluid it experiences the upward thrust and floats.

How Parachute works?
The parachute is made of thin material that traps the air and hence makes it spreads out over larger area. That give out the upward thrust to parachute to fly in air.

Solved Examples

Question 1: A stone weighs 500 N in air and 450 N in water. Calculate the upthrust on the stone.
Solution:
 
Given:
Weight of stone in air = 500 N
Weight of stone in water = 450 N
Therefore, Loss of weight in water = (500 - 450)N
But upthrust = Loss of weight = 50 N
 

Question 2: A body weighs 70 N when fully immersed in water diplaces 2 Litres. How much will it weigh in water?
Solution:
 
A body displaces 2 Litres of water. Since the mass of 1 Litre of water is 1 kg, the mass of 2 kg of water = 2 kg.
Loss in the weight of body in water = Weight of water displaced = Weight of 2 kg of water
= 2 kg $\times$ g = 2 $\times$ 9.8 ms-2 = 19.6 N
Therefore, Weight of body in water = Wt of body in air - loss of weight
= 70 N - 19.6 N = 50.6 N