Making Logically Simple things mathematically complicated:

There are few physical laws that form the bedrock for engineering mechanics and design. One of which is the famous Pascal Law established by Blaise Pascal, a French mathematician in 1653.


The law states that “Pressure at any point in a fluid at rest has the same magnitude in all directions.”

This is one of the most fundamental laws in Engineering Mechanics. I don’t know how Pascal derived this law. But below I have shown how this law is derived in engineering text books and in engineering classes. Later I will show how simple is the law and how the engineering professors and textbooks have complicated the derivation. First read on how it is derived in engineering classes.

Consider an infinitesimal wedge shaped element of fluid at rest as free body. The element is arbitrarily chosen and has dimension as shown below.


Since the fluid is at rest there can be no shear forces, the only forces acting on the free body are the normal pressure forces exerted by surrounding fluid on the plane surfaces, and the weight of the element.
As the element is in equilibrium, the sum of force components on the element in any direction must be equal to zero. (Px, Py and Ps are normal pressures acting on the three faces of the wedge as shown and W is the specific weight of the fluid acting vertically downwards at the Centre of gravity of the wedge)
Therefore, Equilibrium equation in X direction is
Px(dy)(dz) – Ps (ds) (dy) Sin a = 0..................(1)

From the above figure, (dz)= (ds) Sin a ;

So the above equation reduces to
Px(dy)(dz) – Ps (dz) (dy) =0 ;

Dividing the equation by (dz) (dy)
Px – Ps = 0

Therefore:
Px =Ps………………………………………………(A)

Similarly, Equilibrium equation in X direction is

Pz(dy)(dx)–Ps (ds)( dy Cos a)–(W/2)(dy)(dx)(ds) = 0.................(2)

From the figure ;
(ds)( Cos a)= (dx)

Therefore the above equation (2) reduces to;

Pz(dy)(dx)–Ps(dx)(dy)–(W/2)(dy)(dx)(ds)=0.....................(3)

Since (W/2)(dy)(dx(ds) is the product of three infinitesimally small quantities ; this term can be ignored; So (3) reduces to

Pz(dy)(dx) – Ps(dx)(dy)= 0

Dividing by (dy)(dx)
Pz – Ps= 0

Therefore: Pz =Ps………………………………………………..(B)

Combining equations (A) and (B);

Pz = Ps = Px………………………………………………………..(C)

This is the standard derivation followed in all engineering reference books. Obviously most students follow the same derivation. (This also forms a standard 10 marks question in every fluid mechanics examination)

As such this does not look complicated. But even this much complication and mathematics is not required to derive this equation.

A basic property of liquid learnt in primary school tells that liquids do not have a shape and it takes the shape of the container. In that case, it is obvious that the liquid contained in the container must exert equal pressure in all directions (be it 3 or 16 or 10000 directions).
Put other way, if the walls of the container containing a liquid are removed, same volume of liquid flows out in every direction implying that the force exerted at any point in the fluid is same in all direction.

Most simply put, as the liquid takes the shape of the container containing it implies that force exerted at any point in the fluid is same in all directions. (This is a one mark question in 5th or 6th grade).

The point here is that, sometimes obvious things are made mathematically complicated by textbook writers and professors. Unnecessary sophistication is burdened on students, in the process also compromising their inquisitive thinking and creativity.