---
title: "Total pressure and centre of pressure derivation"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:total-pressure-centre-pressure"
source: https://www.bctengineer.com/q/semester_6/hydro_power/total-pressure-centre-pressure
site: BCT Engineering
---

# Total pressure and centre of pressure derivation

Hydro Power · Chapter 1: Introduction to Basic Fluid Mechanics and Hydraulics

- Asked in: 2082 Kartik, 2+4 marks; 2075 Baisakh, 2+3+3 marks (asked 2 times)
- Video: [Total pressure & centre of pressure on vertically immersed surface | Fluid Mechanics | Hydraulics](https://www.youtube.com/watch?v=4iZph-F2wVg) — Professor Nidhi Trivedi
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/total-pressure-centre-pressure
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

Define the terms total pressure and centre of pressure. Derive expressions for total pressure and centre of pressure for a vertically immersed surface.

## Short answer

Total pressure is the resultant force of a static fluid on a surface, given by F = ρ g A h_bar with h_bar the centroid depth. Centre of pressure is the point where this force acts, at depth h* = h_bar + I_G/(A h_bar) below the free surface, always below the centroid.

## Answer

**Total pressure** is the resultant force exerted by a static fluid on a surface in contact with it; it acts normal to the surface. **Centre of pressure** is the point on the surface through which this resultant force acts.

#### Setup
Consider a plane surface of any shape, area $A$, immersed vertically in a liquid of specific weight $\rho g$. Let $\bar h$ be the depth of its centroid G and $h^*$ the depth of the centre of pressure P, both measured from the free surface.

```text
  free surface
 ---------------------------------
      |          |
      | h-bar    | h*
      v          |
  +---G------+   |
  |  [strip] |<- dh at depth h
  |       P  |   v
  +----------+
```

#### Total pressure
Take a thin horizontal strip of width $b$ and thickness $dh$ at depth $h$. Pressure on it is $p = \rho g h$, so
$$
dF = \rho g h\, (b\, dh) = \rho g h\, dA
$$
$$
F = \int \rho g h\, dA = \rho g \int h\, dA = \rho g A \bar h
$$
since $\int h\, dA = A\bar h$ (first moment of area about the free surface).

#### Centre of pressure
Take moments about the free surface line. The moment of $dF$ is $dF\cdot h = \rho g h^2 dA$. Sum of moments equals the moment of the resultant:
$$
F\, h^* = \rho g \int h^2 dA = \rho g\, I_o
$$
where $I_o$ is the second moment of area about the free surface. By the parallel axis theorem $I_o = I_G + A\bar h^2$. So
$$
\begin{aligned}
h^* &= \frac{\rho g (I_G + A\bar h^2)}{\rho g A \bar h} \\
&= \bar h + \frac{I_G}{A\bar h}
\end{aligned}
$$

#### Results
- $F = \rho g A \bar h$ (independent of the shape; depends only on area and centroid depth).
- $h^* = \bar h + \dfrac{I_G}{A\bar h}$. Because $I_G/(A\bar h)$ is always positive, the centre of pressure lies below the centroid; the gap reduces as depth increases.
- Example: rectangle $b \times d$ with top at the surface: $\bar h = d/2$, $I_G = bd^3/12$, so $h^* = 2d/3$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the centre of pressure?
   A. The centroid of the immersed surface
   B. The point on the surface through which the resultant hydrostatic force acts
   C. The point of maximum pressure on the free surface
   D. The point where pressure is zero
   Answer: B. Total pressure is the resultant force on the surface; the centre of pressure is its point of application. (Key point: Definitions)

2. Total pressure on a vertically immersed surface of area A with centroid at depth h-bar is:
   A. rho g A h-bar
   B. rho g h-bar / A
   C. rho g A^2 h-bar
   D. rho g I_G / h-bar
   Answer: A. F = rho g integral(h dA) = rho g A h-bar, since integral(h dA) is the first moment of area about the free surface. (Key point: Total pressure expression)

3. Which step gives the centre of pressure h*?
   A. Equating the force F to rho g I_o
   B. Equating the moment of F about the free surface to the sum of moments of dF, giving F h* = rho g I_o
   C. Differentiating F with respect to depth
   D. Setting h* equal to the centroid depth
   Answer: B. Moments about the free surface give F h* = rho g integral(h^2 dA) = rho g I_o, then the parallel-axis theorem is used. (Key point: Centre of pressure derivation)

4. For a rectangle b x d with top edge at the free surface, the centre of pressure is at depth:
   A. d/2
   B. d/3
   C. 2d/3
   D. 3d/4
   Answer: C. h* = d/2 + (bd^3/12)/(bd x d/2) = d/2 + d/6 = 2d/3. (Key point: Result)

## Related in Hydro Power

- [Centre of pressure below centroid proof](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Acentre-pressure-below-centroid) — Hydro Power, chapter 1
- [Pressure on rectangular plate 3 m × 5 m](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Arectangular-plate-3x5) — Hydro Power, chapter 1
- [Pressure on isosceles triangular plate](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Atriangular-plate-pressure) — Hydro Power, chapter 1
- [Pressure on circular plate (2.5 m)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Acircular-plate-2-5m) — Hydro Power, chapter 1, asked 2×
- [Pressure on circular plate (2.6 m)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Acircular-plate-2-6m) — Hydro Power, chapter 1, asked 2×
- [Pressure on circular plate (2.2 m)](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Acircular-plate-2-2m) — Hydro Power, chapter 1

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