Short Answer:
The hydrostatic law states that the rate of increase of pressure in a vertically downward direction in a static fluid is equal to the weight of the fluid column per unit volume. In simple terms, as the depth of a fluid increases, the pressure increases proportionally due to the weight of the fluid above that point.
This law helps to determine the pressure at any depth in a liquid at rest. It is an important principle in fluid mechanics and is widely used in the design of dams, tanks, submarines, and hydraulic systems to calculate fluid pressure variations with depth.
Detailed Explanation :
Hydrostatic Law
The hydrostatic law is a fundamental principle of fluid statics that explains how pressure varies with depth in a stationary fluid. It is based on the fact that in a fluid at rest, the pressure increases as we move downward due to the weight of the fluid column above.
In a static fluid, there are no shear stresses because the fluid is not moving. Therefore, only normal stresses (pressure) act on any surface within the fluid. The deeper the point within the fluid, the greater the pressure exerted at that point. This is why the bottom of a water tank or dam wall experiences more pressure than the top.
Statement of Hydrostatic Law
The hydrostatic law can be stated as:
“The rate of increase of pressure in a vertically downward direction in a static fluid is equal to the specific weight of the fluid.”
Mathematically, it is expressed as:
Where,
- dP = increase in pressure,
- dh = increase in depth,
- ρ = density of the fluid (kg/m³),
- g = acceleration due to gravity (9.81 m/s²).
This relation shows that pressure in a fluid increases linearly with depth.
Derivation of Hydrostatic Law
To derive the hydrostatic law, consider a small fluid element in the shape of a vertical cylinder of height dh and cross-sectional area A, as shown in fluid mechanics.
Let,
- Pressure on the top face = P,
- Pressure on the bottom face = P + dP,
- Density of the fluid = ρ,
- Acceleration due to gravity = g.
Now, the forces acting on the fluid element are:
- Upward force on the bottom face:
- Downward force on the top face:
- Weight of the fluid element:
Since the fluid is at rest, the element is in equilibrium. Therefore,
Simplifying,
This shows that the change in pressure with depth (dh) is directly proportional to the product of the fluid density (ρ) and gravity (g).
Integrating both sides from the surface (where pressure = P₀) to any depth (h):
Thus, the pressure at any depth (h) below the free surface is given by:
Where,
- P = Pressure at depth h,
- P₀ = Pressure on the free surface (usually atmospheric pressure),
- ρ = Density of the fluid,
- g = Acceleration due to gravity,
- h = Depth below the surface.
This is the mathematical form of the hydrostatic law.
Physical Meaning of Hydrostatic Law
The hydrostatic law explains that pressure in a fluid at rest increases linearly with depth. This happens because of the increasing weight of the fluid above the point considered. The pressure acts equally in all directions (as per Pascal’s law) and is always perpendicular to any surface.
For example, in a water tank:
- At the surface, the pressure equals the atmospheric pressure.
- At 5 m depth, the pressure is .
- At 10 m depth, the pressure doubles compared to 5 m because of the doubled fluid height.
This linear relationship between depth and pressure is crucial for engineers to calculate pressure forces on submerged surfaces and design safe structures.
Key Points from Hydrostatic Law
- Pressure in a liquid increases proportionally with depth.
- The increase in pressure does not depend on the shape or volume of the container — it depends only on the depth and density of the liquid.
- Pressure acts equally in all directions at a given depth.
- The law applies only to fluids at rest and incompressible fluids (liquids).
- It forms the basis for important principles like Pascal’s law, Archimedes’ principle, and buoyancy calculations.
Applications of Hydrostatic Law
The hydrostatic law is used in several practical and engineering applications:
- Design of Dams and Reservoirs:
The pressure on a dam increases with depth, so the dam must be thicker at the bottom than at the top to withstand this pressure. - Submarine and Ship Design:
It helps calculate pressure on submarine hulls and determine how structures behave under deep-sea pressures. - Calculation of Fluid Pressure in Tanks:
Used to determine the pressure at different levels in storage tanks, pipelines, and water supply systems. - Hydraulic Systems:
The pressure transmission in hydraulic systems follows both Pascal’s and hydrostatic laws. - Barometers and Manometers:
Used to measure atmospheric and fluid pressures based on the height of the fluid column. - Buoyancy and Floating Bodies:
The hydrostatic law helps in determining the buoyant force acting on a submerged or floating body, as the force depends on pressure variation with depth.
Example
If a tank is filled with water (ρ = 1000 kg/m³) up to a height of 4 m, the pressure at the bottom is:
This means the pressure at the bottom is 39.24 kPa greater than the atmospheric pressure.
Importance of Hydrostatic Law in Engineering
- Provides a way to calculate pressure at any point in a static fluid.
- Forms the basis for designing fluid containers, dams, submarines, and hydraulic devices.
- Helps understand how pressure forces act on surfaces submerged in fluids.
- Essential for civil, marine, and mechanical engineering applications involving static liquids.
Conclusion
In conclusion, the hydrostatic law states that the pressure in a fluid increases with depth due to the weight of the fluid above. It is mathematically expressed as , and the pressure at any depth is . This law forms the foundation for many important principles and applications in fluid mechanics. It is crucial in the design of hydraulic structures, measuring devices, and submerged systems, ensuring the safe and effective handling of fluids in engineering.