Pascal's Law

 Statement - Pascal's law states that if the fluid is at rest, the intensity of pressure at a point will be same in all directions.

Proof -  Consider a small wedge shaped element in a fluid is at rest.

The forces acting on the element are:

(1) Weight of the element acting vertically downward.

(2) Pressure forces acting normal to the surface

Let,

Px = pressure acting on face AB

Py = pressure acting on face AC

PS = pressure acting on face BC

w = specific weight of the liquid

Force on face Ab = Px X area of face AB = Px X   ( dy X 1) = Pxdy

Force on face AC = Py X area of face AC = Px X   ( dy X 1) = Pydx

Force on face BC = Ps X area of face BC = Px X   ( ds X 1) = Psds

Weight of fluid element = specific weight X volume 

W = w X (area of triangular element X depth) 

W = w X 1/2 (dx X dy X 1)

W = 1/2 w dxdy

As the fluid element is in equilibrium, 

 ΣFx = 0 and  ΣFy = 0

 ΣFx = Algebraic sum of all the forces acting in x- direction = 0

So, Pxdy - Psds sinθ = 0

Pxdy  = Psds sinθ ( Psds sinθ is the horizontal component of force on face BC) 

Pxdy = Psdy  ( ∵ dy = ds sinθ) 

So, Px = Ps                                        ( equation 1)

ΣFy = algebraic sum of all forces along Y direction = 0

Or, Pydx - 1/2 w dxdy - Psds. Cosθ = 0   ( equation 2)

If the size of the element approaches smaller and smaller dimensions then the term

[ 1/2w dxdy]  will be zero. 

So,

 Pydx = Psds. Cosθ

Pydx = Psdx.                                           (∵  dsCosθ = dx) 

Py = Ps                                                      ( equation 3)

On comparing equation 1 and 2

Px = Py = Ps                                     ( equation 4)

As equation 4 is independent of θ which is an arbitrary angle, pressure is same in all direction in stationary fluid.