---
title: "Force to hold hinged rectangular gate"
subject: "Hydro Power"
chapter: 1
key: "semester_6/hydro_power:hinged-rectangular-gate"
source: https://www.bctengineer.com/q/semester_6/hydro_power/hinged-rectangular-gate
site: BCT Engineering
---

# Force to hold hinged rectangular gate

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

- Asked in: 2074 Bhadra, 8 marks
- Video: [Submerged Gate: Moment at Hinge](https://www.youtube.com/watch?v=WkM1tCZDM2c) — LearnChemE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/hydro_power/hinged-rectangular-gate
- Whole chapter: https://www.bctengineer.com/md/semester_6/hydro_power/chapter-1

## Question

A vertical rectangular gate 4 m × 2 m (4 m side being vertical) is hinged at a point 10 cm below the centre of gravity of the gate. The depth of water is 6 m above the bottom of the gate. What is horizontal force must be applied at the bottom of the gate to keep it in vertical position?

## Short answer

Vertical gate (4 m × 2 m) with water 6 m above bottom has water force 313.92 kN at 4.333 m depth. Hinge 0.1 m below centroid (4.1 m depth). Moment balance gives required bottom force 38.55 kN towards water.

## Answer

Find the water force and its line of action, then take moments about the hinge.

**Given:** gate 2 m wide, 4 m high (vertical), water 6 m deep above the bottom of the gate, hinge 0.1 m below the centroid G.

```text
 free surface ============
          | 2 m
     +----------+  top (2 m depth)
     |          |
     |    G     |  4 m   (G at 4 m depth)
     |  hinge o |  4.1 m depth
     |     P    |  4.333 m depth
     |          |
     +----------+  bottom (6 m depth)
              <-- P_b (applied force)
```

#### Water force
$$
\begin{aligned}
A &= 4 \times 2 = 8\ \text{m}^2, \quad \bar h = 2 + 2 = 4\ \text{m} \\
F &= \rho g A\bar h = 9810 \times 8 \times 4 = 313\,920\ \text{N} = 313.92\ \text{kN}
\end{aligned}
$$

#### Centre of pressure
$$
\begin{aligned}
I_G &= \frac{2 \times 4^3}{12} = 10.667\ \text{m}^4 \\
h^* &= 4 + \frac{10.667}{8 \times 4} = 4 + 0.3333 = 4.333\ \text{m}
\end{aligned}
$$

#### Moments about the hinge
- Hinge depth $= 4 + 0.1 = 4.1$ m.
- Water force acts $4.333 - 4.1 = 0.2333$ m **below** the hinge, so it tends to turn the gate open at the bottom.
- Force $P_b$ at the bottom acts $6 - 4.1 = 1.9$ m below the hinge, opposite to the water force.

$$
\begin{aligned}
P_b \times 1.9 &= 313.92 \times 0.2333 \\
P_b &= \frac{73.25}{1.9} = 38.55\ \text{kN}
\end{aligned}
$$

**Answer:** a horizontal force of about **38.55 kN** must be applied at the bottom of the gate, towards the water, to keep it vertical.

## Check yourself

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

1. Gate is 4 m high with water 6 m above its bottom. What is the centroid depth?
   A. 5 m
   B. 2 m
   C. 4 m
   D. 6 m
   Answer: C. Top is at 2 m depth, so G is at 2 + 2 = 4 m. (Key point: Depth of centroid)

2. What is the water force on the 4 m x 2 m gate?
   A. 78.48 kN
   B. 313.92 kN
   C. 470.88 kN
   D. 156.96 kN
   Answer: B. F = 9810 x 8 x 4 = 313.92 kN. (Key point: Water force)

3. The centre of pressure is at 4.333 m depth and the hinge at 4.1 m depth. What does this mean?
   A. Water force acts 0.233 m below the hinge, tending to open the gate at the bottom
   B. Water force acts at the hinge so no moment arises
   C. The gate needs no external force
   D. Water force acts above the hinge
   Answer: A. The force acts below the hinge, so it produces a turning moment about the hinge. This is balanced by the applied force at the bottom. (Key point: Centre of pressure and hinge)

4. Which moment balance gives the force P at the bottom?
   A. P x 1.9 = 313.92 x 0.2333
   B. P x 2 = 313.92 x 0.3333
   C. P x 4 = 313.92 x 4.1
   D. P x 1.9 = 313.92 x 4.333
   Answer: A. Bottom is 6 - 4.1 = 1.9 m below the hinge; the water force is 0.2333 m below it. P = 73.25/1.9 = 38.55 kN. (Key point: Moment equation)

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