Name: A Formal Proof of Archimedes' Principle
Uploaded: 2019-07-19
Duration: 4 min 58 s
Description: Is Archimedes' principle just experimental, or can it be proved mathematically, as well? Watch the video to understand a mathematical proof of Archimedes' pr...

Question

State and prove Pascal&rsquo;s law of fluid pressure. Illustrate it by two examples.

Accepted Answer

According to Pascal's law, pressure in a fluid in equilibrium is the same everywhere, if the effect of gravity can be neglected.

Let us imagine a right circular cylinder in the liquid with the line $C D$ as axis and points $C$ and $D$ as centres of the flat end faces of the cylinder. Let us consider the liquid inside the cylinder which is in equilibrium under the action of forces exerted by the liquid outside the cylinder. These forces are perpendicular to the surfaces of the cylinder everywhere, as shown. It is clear that the forces $F_{1}$ and $F_{2}$ on the flat faces at $C$ and $D$ are perpendicular to the forces on the curved surface of the cylinder. Since the liquid is in equilibrium, the sum of all forces acting on the cylinder must be zero. As such,

$F_{1} = F_{2}$

If $p_{1}$ and $p_{2}$ be the pressures on the flat faces at $C$ and $D$ , and $A$ be the area of cross-section of each face, then,

$F_{1} = p_{1} A$ and $F_{2} = p_{2} A$ or

$p_{1} = p_{2}$

Thus, Pascal’s law is proved. A similar argument can be made to show that the pressure at any point does not depend on the direction of the surface on which it acts.

To illustrate it, we take a flask made of glass, having small tubes jetting out from the sides as well as from the bottom of the flask.

The flask is filled with coloured water. The flask is also fitted with a piston, which can be pushed down. When the piston is pushed down, the water rises up in all tubes to the same height. This shows that pressure in a liquid is equally transmitted in all directions.
Similarly, we take a hollow rubber ball and make on its surface a number of pin-holes at different places. Now, we fill up the rubber ball with water through a wider hole. If, after closing this wider hole, we squeeze the ball, then water comes out of the pin-holes in the form of fine streams. These streams of water rise up to the same height, indicating that pressure applied on water is transmitted equally in all directions.

State and prove Pascal’s law of fluid pressure. Illustrate it by two examples.

Important Questions on Fluid Pressure

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