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Pascal’s Law

Statement: “In n static fluid the pressure or intensity of pressure at a point is equal in all directions”.

Proof: Let us take a fluid element as shown in Fig. 2.1. BC = dx, AB = d y, AE = d z AC = ds._

P x = Pressure on ABOE surface

P y = Pressure on BCDO surface

P θ= Pressure on ACDE surface

Force on                      ABOE = P x d y d z

Force on                      BCDO = PY dx d z

Force on                      ACDE = P θ d z ds.

As the fluid element is in equilibrium

 

Σ F x = 0

or         p x d y d z – p θ d z d S cos e = 0

or                         P x d y d z – Pθ d z d y =0 [ds cos θ=d y]

or                                                    P x = Pθ

And                                  Σ FY =0 [All forces on y direction is zero]

Or               P y dx d z – P θ d z ds sin θ = 0

or                           P y dx d z – Pθ dz dx = 0 [ds sin θ = dx]

or                                                         P y =P θ

thus                                         P x = PY = Pθ (2.1)

 

Equation (2.1) is independent of e, therefore it is proved that pressure acts equally in all directions in a stationary fluid.