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Archimedes Principle

When a body is wholly or partially immersed in a fluid, it displaces the fluid. The displaced fluid exerts an upward force on the body. This tendency of the displaced fluid to exert an upward force on the body immersed in it is called buoyancy. The upward force is called buoyant force or upthrust. This upthrust is clearly equal to the apparent loss in weight. This fact was first discovered by Archimedes, the Greek philosopher. It is known as Archimedes’ principle. It may be stated as follows:-

If a body is wholly or partially immersed in a fluid, it experiences an upthrust equal to the weight of the fluid displaced and this upthrust acts through the center of gravity of the displaced fluid.

The principle has been stated for a fluid, since it holds equally well for gases and for liquids.

Proof of Archimede's Principle

Let us consider a solid cylinder of height h and cross-sectional area A completely immersed in a fluid of constant density p [Fig.1718). The horizontal thrusts on the cylinder balance each other because these are equal in magnitude and opposite in direction. Let us now consider the vertical thrust i.e. thrust on the two end faces of the cylinder.

Suppose the top face of the cylinder is at a depth h1 below the free surface of the fluid. Let P be the atmospheric pressure.

Total downward thrust on the top face of the cylinder = (P + h1pg)A
Total downward thrust on the bottom face of the cylinder = (P + h2pg)A
Here, h2 is the depth of the bottom face below the free surface of the fluid.
Resultant upthrust = (P+h2pg)A - (P+h1pg)A = (h2-h1)pgA
= hpgA [ h2 - h1 = h]

But hA = Volume V of the cylinder
                = Volume of fluid displaced by the cylinder
hAp = Vp = M (mass of fluid displaced)
Upthrust = Mg
Upthrust = Weight of fluid displaced.

A General Discussion of Archimedes Principle and Buoyancy

Let us consider an object of arbitrary shape immersed in a liquid [Fig.17.19(a)]
The surrounding liquid exerts pressure at all points on the surface of the object and we wish to calculate the resulting force that acts upwards on it. That the resultant force must act upwards is clear, since the pressure increases with depth, so that the force must be greater on the underside of the object than on the upper side

Now imagine that the irregular object is replaced by the same liquid. This liquid will occupy precisely the same space as was previously occupied by the solid, as indicated in Fig.17.19(b). Since the replacement liquid has the same boundary as the solid object had, the remainder of the liquid must exert upon it precisely the same forces as it exerted on the solid object. But the lump of liquid that has replaced the solid is at rest, and therefore there is no unbalanced force acting on it and no unbalanced torque acting on it. The weight of the lump of liquid acts downwards through its center of gravity, and consequently we deduce that the rest of the liquid exerts an upward force on it equal to its weight and that is upward force acts through its center of gravity. Hence, when the solid was there, the liquid exerted upon it an upward force equal to the weight of the liquid displaced, this force acting through the center of gravity of the displaced liquid. It will be noticed that no mathematics is needed at all for this proof even although we have considered a body of quite an arbitrary shape. Legend has it that this beautiful piece of logical analysis occurred to Archimedes in his bath, and that he was so exhilarated by it that he leaped up and ran naked through the streets to tell people about it.

To put the principle into more formal terms, suppose the immersed object has a volume V and is of density p; then its weight is Vpg = W. If the density of the liquid is p’ < p, the weight of liquid displaced is given by W’ and v’g, since a solid object must obviously displace its own volume of liquid. Hence, the resultant downward force, or effective weight is
W" = W - W' = V(p-p')g = Vpg = W

This correction to the weight of an object must be must be taken into account in high precision work, even when the fluid is air.

If p'>p, then the body will rise to the surface of the liquid under the action of the resultant upward force, and it will settle in equilibrium when just so much of it is immersed as to give a buoyancy upthrust equal to its own weight. When p’=p, there will be no resultant force acting on the body, and it will remain at rest at any height completely immersed in the liquid. This is a most convenient method of measuring density of very small objects. The procedure is to mix two liquids of different densities until the density of the mixture is just equal to that of the unknown density of the object. The point of equality is observed, as the object shows no tendency to either rise or sink.

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