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.