Short Answer:
The pressure at a point in a static fluid is the same in all directions and acts perpendicularly (normally) to any surface passing through that point. It depends only on the depth of the fluid, its density, and the acceleration due to gravity.
In simple words, in a fluid at rest, pressure does not have any direction of preference—it is uniform in every direction at a given point. The pressure increases with the depth in the fluid because of the weight of the fluid above that point. Mathematically, it is given by P = P₀ + ρgh, where ρ is the density and h is the depth.
Detailed Explanation :
Pressure at a Point in a Static Fluid
The concept of pressure at a point in a static fluid is fundamental in the study of fluid mechanics. It helps in understanding how fluids behave when they are at rest and how they exert forces on the boundaries or surfaces in contact with them.
When a fluid is stationary, there are no shear forces acting within it. The only type of stress that exists is normal stress, which is the same in every direction. This normal stress per unit area is called pressure. Hence, the pressure at a given point in a fluid is always the same in all directions and acts perpendicular to the surface at that point.
This property makes fluids unique and forms the basis for important fluid mechanics laws such as Pascal’s law and the hydrostatic law.
Explanation of Pressure at a Point
To understand pressure at a point, consider a small fluid element located at some depth in a container filled with a liquid, such as water. The fluid is at rest, which means no motion occurs within it.
Now, let us imagine an infinitesimally small cube of fluid around the point of interest. The cube has surfaces perpendicular to three coordinate axes (x, y, and z). The pressure acting on each face of the cube is perpendicular to that face.
Let:
- = pressure on the face normal to the x-direction
- = pressure on the face normal to the y-direction
- = pressure on the face normal to the z-direction
For the fluid to be in equilibrium (since it is at rest), the pressure acting on all faces of the cube must be equal, that is:
This shows that pressure at a point in a static fluid is the same in all directions.
Mathematical Derivation
To prove that pressure at a point in a static fluid is equal in all directions, consider a very small triangular prism of fluid inside the fluid mass.
Let the three faces of the prism be:
- A horizontal face (AB),
- A vertical face (AC), and
- An inclined face (BC).
Let the pressures acting on these faces be , , and respectively.
Let the area of the inclined face BC be .
Then:
- Area of the vertical face AC =
- Area of the horizontal face AB =
The forces acting on the prism are:
- Pressure forces on each face,
- The weight of the fluid element (W = ρg × Volume).
For equilibrium in the vertical direction:
and for equilibrium in the horizontal direction:
Solving these equations, we get:
Thus, the pressure at a point in a static fluid is the same in all directions. This property is also known as isotropy of pressure.
Formula for Pressure at a Point
The pressure at a point at depth h below the free surface of a static fluid is given by the hydrostatic equation:
Where,
- = pressure at depth h (N/m²),
- = pressure at the free surface (N/m²),
- = density of the fluid (kg/m³),
- = acceleration due to gravity (9.81 m/s²),
- = depth below the surface (m).
If the fluid surface is open to the atmosphere, is the atmospheric pressure, and the total pressure at depth h becomes:
If we only consider the pressure due to the fluid column (excluding atmospheric pressure), it is called gauge pressure:
Physical Meaning
The physical meaning of this concept is simple:
- The deeper you go in a fluid, the greater the pressure.
- The pressure acts equally in all directions at a point, which means there is no shear stress in a fluid at rest.
- Pressure always acts perpendicular to any surface (normal direction).
This uniform behavior of pressure in all directions ensures that structures like dams, tanks, and submarines experience balanced pressure forces from the surrounding water.
Example
Consider a water tank open to the atmosphere.
Let the density of water and depth .
The atmospheric pressure at the surface .
Then,
Thus, the pressure at a point 5 meters below the surface is 150.4 kPa (absolute pressure).
Applications in Engineering
- Design of Dams and Reservoirs:
Engineers calculate pressure variation with depth using this concept to ensure dams are designed thicker at the bottom. - Hydraulic Systems:
Pressure equality at a point ensures uniform force transmission in hydraulic machines. - Submarine and Ship Design:
The pressure at a point determines the stress acting on the hull of a submarine at different depths. - Fluid Measurement Devices:
Pressure at a point is used in barometers, piezometers, and manometers to measure fluid pressures. - Buoyancy Calculations:
Pressure variation with depth helps determine buoyant force on submerged or floating bodies.
Key Points
- Pressure at a point in a static fluid acts equally in all directions.
- It is a scalar quantity and has no specific direction.
- Pressure always acts normal to the surface in contact.
- It increases linearly with depth due to the weight of the fluid above.
- There are no shear stresses in static fluids; only normal stresses (pressure) exist.
Conclusion
In conclusion, the pressure at a point in a static fluid is the same in all directions and acts normally to the surface. It depends on the fluid density, gravitational acceleration, and depth of the point below the surface. This principle forms the foundation of fluid statics and is essential for analyzing fluid pressures in engineering applications such as dam design, hydraulic machines, and pressure measuring instruments. Pressure uniformity at a point ensures that static fluids maintain equilibrium and transmit force efficiently in all directions.