Mechanical Properties of Fluids

Chapter-2  Mechanical Properties of Fluids

Variation of pressure with depth (or due to a liquid column or gauge pressure)-

Consider a liquid of density ρ   is at rest in a Container. Suppose an imaginary cylinder of liquid column of height H of cross-sectional area A.

let P₁ and P₂ be the liquid pressure on upward and lower portion of the cylinder as shown in fig.

Mass of cylindrical liquid column

                             m=ρv

                   m=ρAH          -(1)

                   weight(w)=mg

                   

                   w=ρAHg         -(2)

 where m= mass of the liquid.

            w= force or weight that are acting on cylindrical body due to gravity.

            H= height of the cylinder.

            g= accelaration due to gravity.

            ρ= Density of  the liquid.

            P= pressure exerted liquid on body.

let F and F forces corresponding to pressure P and P

                 F₁=P₁A            -(3)

                 F₂=P₂A            -(4)

 Since liquid at rest

                F₁ + W=F₂

                F₂ - F₁=w

                F₂ - F₁=ρAHg

                From 3 and 4

                P₂A - P₁A=ρAHg

                (P₂ - P₁)A=ρAHg

                 P₂ - P₁=ρHg

Variation in liquid pressure depends only on height not on the shape of area.

Case1- If the first point of the cylinder is on the upper surface of the liquid  then P₁=0 (In closed vessel)

 then P=P₂=ρHg

 This pressure is known as Gauge pressure.

Case2- If the vessel is open then atomsphere(P。) will be also there then the total pressure at a depth H at a liquid will be P=P。+ρHg .

Hydrostatic Paradox-

The liquid pressure depend only on depth and not on the area or same of the container. In other word on the same horizontal level of the liquid ( if the density is uniform ) pressure will be same this is known as hydrostatic paradox.