Hydrostatic force is the force exerted by the static fluid on any object placed into it. It depends on the depth of the object from the free surface. Hydrostatic force for vertical and inclined surfaces will be different for horizontal surfaces. This topic comes under the fluid statics part.
Fundamental concepts in fluid mechanics include calculating the hydrostatic force and the location of the centre of pressure. A location on the submerged surface where the hydrostatic pressure acts is known as the centre of pressure.
The resultant force created by a liquid’s pressure loading acting on submerged surfaces is known as hydrostatic force.
When a surface is immersed in a fluid, the fluid’s forces act on the surface. These forces must be determined for designing storage tanks, ships, dams, and other hydraulic structures. Since there are no shearing stresses in place for fluids at rest, we know that the force must be perpendicular to the surface. If the fluid is incompressible, the pressure changes linearly with depth.
Hydrostatic forces result from a liquid’s pressure loading acting on submerged surfaces. The total hydrostatic force for a horizontal plane surface submerged in liquid, a plane surface inside of a gas chamber, or any other plane surface subject to the influence of uniform hydrostatic pressure is given by:
F = pA
where A is the area and p is the uniform pressure.
Pascal’s Law for Pressure At A Point
It states that pressure or intensity of pressure at a point in a static fluid (fluid is at rest) is equal in all directions. If the fluid is not in motion, then according to Pascal’s law,
px = py = pz
where px, py and pz are the pressure at points x,y, and z, respectively.
According to Pascal’s law, fluid pressure will be constant at all points of a horizontal surface. The variation of hydrostatic force will occur for the inclined surface and for the vertical surface. And this variation will happen along the depth of the surface from the free surface.
Taking upward as a positive, we have
Vertical cylindrical element of fluid cross-sectional area = A
mass density = ρ
The forces involved are:
= mass density x volume x g
= ρ.g.A.(z2 – z1)
Thus in a fluid under gravity, pressure decreases linearly with an increase in height.
p2 – p1 = ρgA(z2 – z1 )
This is the hydrostatic pressure change.
As we know Pascal’s law, so it can be understood that the hydrostatic pressure at the horizontal surface is the same at all points. And pressure intensity will be uniform.
Horizontal cylindrical element cross-sectional area = A
mass density =ρ
left end pressure = pl
right end pressure = pr
For equilibrium, the sum of the forces in the x-direction is zero= pl. A = pr. A
pl = pr
So, Pressure in the horizontal direction is constant.
The hydrostatic force depends on the type of surface, whether horizontal, vertical or inclined. Here the calculation of hydrostatic forces for different types of surfaces and flow conditions are explained below.
F = p. A
where p is the uniform pressure and A is the area.
F = pcg. A
where pcg is the pressure at the centre of gravity.
F=γh’A
Derivation of Formulas
The figure shown below is an inclined plane surface submerged in a liquid. The total area of the plane surface is given by A, cg is the centre of gravity, and cp is the centre of pressure.
(Forces on an inclined plane surface)
The differential force dF acting on the element dA is
dF=p. dA
dF=γ. h. dA
From the figure
h=ysinθ,
dF=γ.(ysinθ). dA
Integrate both sides and note that γ and θ are constants,
F=γ. sinθ. ∫y.dA
So, F=γ. sinθ. ∫y.dA
Recall from Calculus that
∫y.dA=A.y¯
Hence, F=(γ.sinθ)A.y¯
F=γ. (y¯sinθ). A
From the figure, y¯sinθ=h¯, thus,
F = γh¯A
The product γh¯¯ is a unit pressure at the centroid at the plane area; thus, the formula can be expressed in a more general term below:
F = pcg. A
Location of Total Hydrostatic Force (Eccentricity)
From the figure above, S is the intersection of the prolongation of the submerged area to the free liquid surface. Taking a moment about point S.
Fyp=∫y. dF
Where
dF=γ(ysinθ)dA
F=γ(y¯sinθ)A
[γ(y¯sinθ)A]yp =∫y[γ(ysinθ)dA][γ(y¯sinθ)A]yp
=∫y[γ(ysinθ)dA]
(γsinθ)Ay¯yp=(γsinθ)∫y2dA(γsinθ)Ay¯yp
=(γsinθ)∫y2dA
Ay¯yp=∫y2dA
Again from Calculus, ∫y2dA is the moment of inertia denoted by I. Since our reference point is S,
Ay¯yp=IS
Thus,
yp =IS/Ay¯
By transfer formula for the moment of inertia IS=Ig+Ay¯2, the formula for yp will become
yp=(Ig+Ay¯2)/Ay¯ or
yp=y¯+Ig/Ay¯
From the figure above, yp=y¯+e; thus, the distance between CG and CP is
Eccentricity, e=Ig/Ay¯
he concept of hydrostatic force has many applications in real life. For example, lifting a hydraulic jack can be easily possible with hydrostatic force; this principle is commonly used at the car washing centre. Here a few applications of hydrostatic force are listed below.
Force on a curved surface can be calculated by splitting it into verticle surfaces and horizontal surfaces. Here the different cases of hydrostatic force on a curved surface are explained.
FH=pcg. A
Vertical Component, The vertical component of the total hydrostatic force on any surface is equal to the weight of either real or imaginary liquid above it.
FV=γ. V
F=√(FH2+FV2)
tanθx=FV/FH
There are various applications of hydrostatic force. This force provides various advantages because of its property. The structural design of water-control structures like dams, floodwalls, and gates is heavily influenced by the position and strength of the water pressure force pressing on those structures.
Many hydraulic equipment components must be designed following the principles of hydrostatic force and its course of action.
The hydrostatic force exerted on the vertical surface of the quadrant when it is submerged by adding water to the tank can be calculated by taking into account the following:
mg × L = F × y
As was discussed, hydrostatic force is very important for the GATE and other competitive exams. So it is important to understand this topic with a better approach. A common example related to this topic is given that strengthens the related concepts.
Example: The length of a tainter gate is 1 m perpendicular to the plane of the paper. Find out the total horizontal force on the gate and hydrostatic force on the gate.
Sol:
1. What is hydrostatic force? |
2. What is the formula for hydrostatic force? |
3. What is the general equation for the variation of pressure in a static fluid? |
4. Why is the pressure equal at the same level in a static fluid? |
5. How does hydrostatic force affect submerged objects? |
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