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# State the law of floatation of bodies in a liquid. Arrive at an expression for the fraction of volume of the floating body submerged in the liquid.

Last updated date: 14th Sep 2024
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Hint: Archimedes has calculated the amount of buoyant force exerted by the liquid on the body. The weight of the liquid displaced will be equivalent to the weight of the object. Using this find the fraction of the volume of the body submerged. This will help you in answering this question.

Archimedes has calculated the amount of buoyant force exerted by the liquid on the body. According to the Archimedes, if a body is completely or partially immersed in a liquid at rest, it will experience an upthrust, identical to the weight of the liquid displaced by the body.
Let us assume that $x$ be the fraction of volume of the body floating above the surface of the liquid. We can write that,
$\text{weight of the liquid displaced}=\text{weight of the object}$
This can be written as,
$\therefore \left( {{V}_{0}}-x{{V}_{0}} \right)dg={{V}_{0}}{{d}_{0}}g$
Where ${{V}_{0}}$ be the volume of the liquid, $d$ be the density of the liquid, ${{d}_{0}}$ be the density of the object and $g$ be the acceleration due to gravity.
The equation can be simplified as,
$\left( 1-x \right)d={{d}_{0}}$
Therefore the fraction of volume of the floating body submerged in the liquid can be found by rearranging this equation,
$x=1-\dfrac{{{d}_{0}}}{d}=\dfrac{d-{{d}_{0}}}{d}$