Hydrostatics - Important Concepts for JEE

Hydrostatic is the branch of fluid mechanics that deals with the equilibrium condition of a floating body and immersed body in the fluid at rest position. In this field, the pressure force exerted by the static fluid on the body which was immersed at a depth from the free surface of the fluid has been studied. The body is immersed in a fluid with different positions and the pressure force acting on the body is analysed. The point of the application of the pressure force acting on the immersed body is called the centre of pressure.

In this article, all concepts of hydrostatics are discussed in the following sections. This article also presents the principle of hydrostatics, the centre of pressure hydrostatics, and the laws of hydrostatics on which hydrostatic forces act on the immersed body. 

Principle of Hydrostatics

The principle of hydrostatics that works on the mechanics of fluid at rest is Pascal’s law. Pascal's law is the fundamental principle that states that the pressure applied to the surface of the fluid is transmitted uniformly throughout the fluid in all directions. The variation in pressure of fluid remains constant. 

Example of pascal's law: when the force is applied on one piston, then the height of the other piston is raised due to the transmission of force through fluid that connects both of them.  

The Pressure of the Fluid at Rest

According to the nature of the fluid, it is difficult for the fluid to remain at rest due to the presence of shear stress. This shear stress develops a shearing action along with the fluid layers which make the fluid spread on the horizontal surface placed in contact with it. But the above condition violates the fundamental principle of hydrostatics. 

When an infinitely small cube of the fluid element is considered in the fluid, then the pressure applied to the fluid element is uniformly distributed and transmitted equally in all directions. The pressure applied on fluid at rest is isotropic. These characteristics make the fluids flow throughout the length of the pipes. Similarly, the pressure force is also transmitted through the length of the pipes and tubes. 

Hydrostatic Pressure

Hydrostatic pressure is the pressure that is acted on the body when it is immersed in water to a certain depth. The Hydrostatic pressure works on the law of hydrostatics which states that,

“The pressure exerted on the immersed object by the fluid is directly proportional to the depth of the object from the free surface of the fluid”

In simple terms, the amount of pressure applied to the object depends upon the depth of the immersed object from the free surface of the fluid. Mathematically,  the equation of hydrostatic pressure is derived in the following manner!

Suppose a body is immersed at a height Z from the free surface of the fluid. For the determination of the hydrostatic pressure on the immersed body, it is necessary to equate the net force acting on the body in the vertical direction. 

Image: This image is the representation of the immersed body in the fluid.

Therefore, 

Net force in downward direction = weight of immersed body + pressure of the liquid on the upper surface

Net force in downward direction = (volume $\times$ density $\times$ acceleration due to gravity) + pressure of the liquid on the upper surface

Net force in a downward direction $=\rho \times g \times A \times d z+p_{z} A$

Net force in an upward direction $=\left(p+\dfrac{d p}{d z} d z\right) \times A$

Therefore, in equating both of them,

$\begin{align} &p_{z} A+\rho \times g \times A \times d z=\left(p+\dfrac{d p}{d z} d z\right) \times A \\ &p_{z} A+\rho \times g \times A \times d z-\left(p_{z}+\dfrac{d p}{d z} d z\right) \times A=0 \\ &p_{z} A+\rho \times g \times A \times d z-p_{z} A-A \dfrac{d p}{d z} d z=0 \\ &\rho \times g \times A \times d z-A \dfrac{d p}{d z} d z=0 \end{align}$

$\begin{align} &\rho \times g \times A \times d z=A \dfrac{d p}{d z} d z \\ &\dfrac{d p}{d z}=\rho g \\ &d p=\rho g d z \end{align}$

On integrating both sides for the evaluation of total pressure, 

$\begin{align} &P=\int d p=\int \rho g d z \\ &P=\rho g Z \end{align}$

Hence, it is verified that the pressure on the immersed body will increase as the depth of the body increases from the free surface of the fluid. 

Centre of pressure hydrostatics:

The Centre of pressure is the point of application of pressure of the fluid acting on the body. If the one end of the body lies on the surface of the body, then the centre of pressure will lie at the geometric centre of the immersed body. 

Conclusion

This article concludes with brief interesting information about the concept of hydrostatics. This article shares a glimpse of the pascal’s law, the law of hydrostatic force for the immersed body at a certain depth from the free surface of the fluid, centre of pressure, etc. It has also discussed the derivation of hydrostatic pressure on the object briefly. This article tried to give a better understanding of the concept of hydrostatic.