Define total pressure and centre of pressure?

Short Exam-Oriented Summary

• Total pressure is the total hydrostatic force exerted by a fluid on an immersed surface.
• Centre of pressure is the point where this force acts.
• For horizontal surfaces, centre of pressure = centre of gravity.
• For vertical, inclined, and curved surfaces, centre of pressure lies below the centre of gravity

Definition of Total Pressure

Total pressure is defined as the resultant force exerted by a static fluid on a surface (plane or curved) when the fluid is in contact with that surface.
This force always acts normal (perpendicular) to the surface.

Mathematically, total pressure depends on:

• the specific weight of the liquid,
• the area of the surface, and
• the depth of the surface below the liquid level.

• Also read about the pressure gauge least count

Definition of Centre of Pressure

The centre of pressure is defined as the point of application of the resultant pressure force on an immersed surface.
It is the point where the entire hydrostatic force can be assumed to act.

The centre of pressure does not coincide with the centre of gravity, except in special cases (such as horizontal surfaces), because pressure increases with depth.

Total Pressure and Centre of Pressure on Immersed Surfaces

1. Horizontally Immersed Surface

For a horizontal surface immersed in a liquid:

Total Pressure (P):$$P = w \times A \times \bar{x}$$P=w×A×xˉ

Where:

• $w$w = specific weight of liquid
• $A$A = area of immersed surface
• $\bar{x}$xˉ = depth of centre of gravity of the surface from liquid surface

✔ This formula is valid for both flat and curved surfaces.

Centre of Pressure:
For a horizontal surface, the centre of pressure coincides with the centre of gravity of the surface.

2. Vertically Immersed Surface

Total Pressure (P):$$P = w \times A \times \bar{x}$$P=w×A×xˉ

Depth of Centre of Pressure (xₚ):$$x_p = \bar{x} + \frac{I_g}{A \bar{x}}$$xp​=xˉ+AxˉIg​​

Where:

• $A$A = area of the surface
• $\bar{x}$xˉ = depth of centre of gravity
• $I_g$Ig​ = moment of inertia of the surface about its horizontal centroidal axis

3. Inclined Immersed Surface

Total Pressure (P):$$P = w \times A \times \bar{x}$$P=w×A×xˉ

Depth of Centre of Pressure:$$x_p = \bar{x} + \frac{I_g \sin^2 \theta}{A \bar{x}}$$xp​=xˉ+AxˉIg​sin2θ​

Where:

• $\theta$θ = angle of inclination of the surface with the liquid surface

4. Curved Immersed Surface

The total force on a curved surface is obtained by resolving it into horizontal and vertical components.

Resultant Force (R):$$R = \sqrt{P_H^2 + P_V^2}$$R=PH2​+PV2​​

Direction of Resultant Force:$$\tan \theta = \frac{P_V}{P_H}$$tanθ=PH​PV​​

Where:

• $P_H$PH​ = horizontal force = total pressure on the vertical projection of the curved surface
• $P_V$PV​ = vertical force = weight of the liquid supported by the curved surface up to the liquid level

Key Difference Between Total Pressure and Centre of Pressure