Archimedes Principle (Buoyance):
It is a common experience that a body appears lighter when immersed in water. Drawing a bucket full of water from a well appears light as long as it is inside water. A boy taking a bath in the river must have observed that his apparent weight decreases as more of his body is submerged in water.
The cause of the decrease in the weight of a body when immersed in any fluid must be some upward force exerted by the fluid on the submerged object.
The upward push applied by the fluid on a submerged object is known as buoyant force and this effect is called buoyancy.
It is because of buoyancy that people can swim, ships can float, and helium-filled balloons can rise through the air. An experimental study of the buoyant force was first carried out by a Greek Scientist Archimedes. After a very careful study, he arrived at a principle known as Archimedes’ Principle.
Archimedes’ Principle states that when a body is wholly or partially immersed in a fluid, it experiences an upward force equal to the weight of the fluid displaced.
If the volume of the body inside the fluid is V, the upward force is given by
| FB = Vρg |
Where ρ is the density of fluid. It is on account of this upward force that a body immersed wholly or partially inside a liquid loses a part of its weight. If the original weight of the body is W, then under liquid its effective weight W’ would be given
| W’ = (W – FB) |
Theoretical Proof:
Let a rectangular block ABCD be immersed in a liquid. The liquid exerts thrust on the vertical sides AD and BC and also on the horizontal surfaces AB and CD.
The thrust due to pressure on the vertical side BC is exactly balanced by an equal and opposite thrust on the opposite AD in the same horizontal line.
The thrust due to liquid pressure on the top surface AB and bottom surface CD act vertically and do not exactly cancel each other. A net thrust, therefore, acts in the upward direction. In order to calculate this upward thrust, let
| density of the liquid = ρ area of cross-section of the block = p1 depth of the upper surface = h1 depth of the lower surface = h2 If the liquid pressure on the upper surface of the block is p1 and on the lower surface is p2, then p1 = h1ρg and p2 = h2ρg Let the thrust on the upper surface of the block be F1 and on the lower surface be F2, then F1 = p1A = Ah1ρg ↓ F2 = p2A = Ah2ρg ↑ Since h2 > h1 a resultant upward thrust FB acts on the block, which is given by FB = F2 – F1 = Ah2ρg – Ah1ρg = A(h2 – h1) ρg Now since A(h2 – h1) is nothing but the volume V of the liquid displaced, we have FB = Vρg which is what Archimedes’ principle states. |
Alternative Proof: Figure (a) shows a body of irregular shape submerged in a liquid. The pressures at various points in the liquid cause various forces to be exerted on the submerged object from all directions. The resultant of all these forces is obviously the upthrust FB that acts on the submerged object due to the surrounding liquid.
In order to find an expression for the upthrust FB, let us imagine that the space occupied by the “submerged object” is filled by the “same liquid” as the rest of the liquid as shown in figure (b). Since the rest of the liquid is exerting forces exactly of the same magnitude and in the same direction as when the submerged object was present, the upward force or upthrust acting on the “irregular mass of the liquid” replacing the submerged object must also be FB.
To calculate FB, we note that the forces acting on the “irregular mass of the liquid” replacing the submerged object are the following:
(1) Upthrust FB due to the rest of the liquid acting upwards.
(2) Force Vρg due to the weight of the “irregular mass of liquid” replacing the submerged object, acting downwards, where V is the volume of an irregular mass of liquid and ρ is its density.
Now, as the “irregular mass of the liquid” is in equilibrium and is floating on the rest of the liquid, the net force on it must be exactly zero. Hence, for equilibrium,
| FB = Vρg |
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