Pascal law states, “The intensity of pressure at a point in a fluid at rest is same in all direction.”


Pascal law: Consider an arbitrary fluid element of wedge shape in a fluid mass at rest as shown in. Let the width of the element be unity.



Let P_x, P_y and P_z be the pressures acting on the face AB, AC and BC respectively.

Let < ABC = ?

Forces acting on the element are

Force on the face AB = pressure in x-direction x Area of face AB
  = P_x (d_y x 1)

Force on the face AC = pressure in y-direction x Area fo face AC
= P_y (d_y x 1)

Force on the face BC = pressure in z-direction x Area of face BC
 = P_z (ds x1)

Weight of the element = V x γ = A x1x γ

= ½ x AC x AB x 1 x γ = dx.dy. γ / 2

Consider the triangle ABC as shown in

cos ? = AB/AC       sin ? = AC/BC

C. cos ? = AB         AC = BC. sin ?

ds. cos ? = dy         dx = ds. sin ?

Resolve the force in x-direction

P_x . d_y x 1 - (P_z cos ?) d_s x 1 = 0

P x d_y = P_z . ds cos ?                          but ds cos ? = dy

P_x. d_y = P_z . d_y

P_x = Pz

Resolve the forces in y-direction

P_y (d_x x1) – P sin ? (d_s x 1) -1/2 d_x . d_y . γ = 0

dx =dx . sin ? and also the element is very small and hence weight is negligible

P_y . d_x-P_z. d_x = 0
    
P_y = P_z

From equation (i) and (ii)

P_x = P_y = P_z

•    Since the choice of fluid element was completely arbitrary, which means the pressure at any point is same in all direction.

•    Pascal’s law states that , “The pressure or intensity of pressure at a point in a static fluid is same in all direction.”

•    It is used for the construction of machines such as hydraulic press, hydraulic riveter etc. In which by the application of relatively small forces considerably larger forces are developed.

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