Now consider a fluid element in the form of a horizontal bar of uniform cross-section. The bar is in equilibrium. The horizontal forces exerted at its two ends must be balanced or the pressure at the two ends should be equal. This proves that for a liquid in equilibrium the pressure is same at all points in a horizontal plane.
Suppose the pressure were not equal in different parts of the fluid, then there would be a flow as the fluid will have some net force acting on it. Hence in the absence of flow the pressure in the fluid must be same everywhere in a horizontal plane. . .
Variation of Pressure with Depth Consider a fluid at rest in a container. In Fig. . point is at height h above a point .
The pressures at points and are P and P respectively. Consider a cylindrical element of fluid having area of base A and height h . As the fluid is at rest the resultant horizontal forces should be zero and the resultant vertical forces should balance the weight of the element. The forces acting in the vertical direction are due to the fluid pressure at the top ( P A ) acting downward, at the bottom ( P A ) acting upward.
If mg is weight of the fluid in the cylinder we have ( P − P ) A = mg ( . ) Now, if ρ is the mass density of the fluid, we have the mass of fluid to be m = ρ V = ρ h A so that P − P = ρ gh ( . ) Fig. .
Proof of Pascal’s law. ABC-DEF is an element of the interior of a fluid at rest. This element is in the form of a right- angled prism. The element is small so that the effect of gravity can be ignored, but it has been enlarged for the sake of clarity.
Fig. . Fluid under gravity. The effect of gravity is illustrated through pressure on a vertical