The principle states that “when a body is partially or completely immersed in a fluid, it experiences a buyont force (upthrust) due to which it appears lighter, the apparent loss in the weight of body is equal to the weight of the liquid displaced”.
Proof of Archimedes’ Principle:
Consider a cubical block ABCDEFGH of face area ‘A’ is completely immersed in a liquid of density p. Let depth of upper face ABCD and that of lower face HEFG with respect to free surface of the liquid is h1and h2 respectively. The liquid exerts pressure on the block from all sides. The thrust due to liquid pressure on the vertical face AHED is exactly balanced by the thrust on the face BCFG. In the same way the trust on the face ABGH is balanced by the thrust due to liquid pressure on vertical face DCFE.
But the thrust due to liquid pressure on the top face ABCD and the bottom face HEGH do not exactly cancel each other as fluid pressure at the lower face is more than the upper face. A net thrust, therefore, acts in the upward direction. Now if the net pressure on the upper face is p1 and on the bottom face, same is, P2.Then, P1 = h1 ρ g
And P2 = h2 ρ g
Further let the thrust on the upper face is F1 and at the lower face it is F2.
Thus, F1 = P1A = h1 ρ g A
and F2 = P2 A= h2 ρ g A
Now since, F2 > F1 (h2 > ht), a resultant upward thrust F is acting on the block, given by,
F = F2 – F1 = h2 ρ g A – h1 ρ g A
F = A (h2 – h1) ρ g
Here in the above expression F = A (h2 – h1) ρ g, the term, A (h2 – h1) is the volume V of the liquid displaced, thus, net up thrust on the solid
F = V ρ g
Or F = weight of the liquid displaced
Therefore, the buyont force (upthrust) on the immersed solid is equal to the weight of the liquid displaced, which is what Archimedes’ principle states.
The Archimedes’ principle has got numerous applications in physics some of them are as under.
⇒ It is used to determine the relative density of a solid and liquid.
⇒ It is used to determine the amount of impurity in a given sample of a body.