---
title: "Resultant force on curved surface ABCDE (S = 1.25)"
subject: "Fluid Mechanics"
chapter: 3
key: "semester_3/fluid_mechanics:curved-surface-abcde-s125-10m"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/curved-surface-abcde-s125-10m
site: BCT Engineering
---

# Resultant force on curved surface ABCDE (S = 1.25)

Fluid Mechanics · Chapter 3: Hydrostatics

- Asked in: 2072 Chaitra, 8 marks
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/curved-surface-abcde-s125-10m
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-3

## Question

Find the resultant pressure force on curved surface ABCDE due to liquid with specific gravity S = 1.25, take length of the curved surface (normal to the paper) as 10 m. [Figure: free surface at A, vertical wall A-B 4 m, then curved portions with 2 m radii marked, vertical dimensions 4 m, 2 m, 2 m, 2 m, curve ends on the ground at E and C, D lowest point]

## Short answer

With γ = 12.26 kN/m³, net horizontal force from the 4 m wall is 981 kN; vertical force is 2732.5 kN. Resultant 2903 kN at 70.3° below horizontal.

## Answer

Resultant force on a curved surface = $\sqrt{F_H^2 + F_V^2}$, where $F_H$ is the force on the vertical projection and $F_V$ is the weight of the liquid vertically above the surface up to the free surface.

**Assumed geometry** (the figure is not fully clear): A is at the free surface; AB is a vertical wall of 4 m; the surface then dips as a semicircular trough of radius 2 m from B to E through the lowest point D, so D is 6 m below the free surface. $S = 1.25$, $L = 10$ m, $\gamma = 1.25\times 9.81 = 12.26$ kN/m³.

```text
 A |~~~~~~~~~~~~~~~~~~~~~~~~ free surface
   | 4 m
 B |                     E
    \         R = 2 m   /
      '--.   C   .--'   
           ' D '          D lowest (6 m)
```

**Horizontal component**

On the trough, the forces on the left and right quarters are equal and opposite horizontally, so they cancel. Only the wall AB (depth 4 m) gives a net horizontal force:

$$
F_H = \gamma\frac{h^2}{2}L = 12.26\times\frac{4^2}{2}\times 10 = 981.0\ \text{kN}
$$

**Vertical component**

Volume above the trough = rectangle of 4 m depth over the 4 m width + half-disc of radius 2 m:

$$
\begin{aligned}
A &= 4\times 4 + \frac{\pi\times 2^2}{2} = 16 + 6.283 = 22.28\ \text{m}^2\\
F_V &= \gamma A L = 12.26\times 22.28\times 10 = 2732.5\ \text{kN (downward)}
\end{aligned}
$$

**Resultant**

$$
\begin{aligned}
F &= \sqrt{981.0^2 + 2732.5^2} = 2903.2\ \text{kN}\\
\theta &= \tan^{-1}\frac{2732.5}{981.0} = 70.3^\circ \text{ below the horizontal}
\end{aligned}
$$

**Answer: $F \approx 2903$ kN, inclined at $70.3^\circ$ to the horizontal; $F_H = 981$ kN, $F_V = 2732.5$ kN.**

## Check yourself

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

1. Why does only the wall AB give a net horizontal force?
   A. Horizontal forces on the two halves of the trough cancel
   B. The trough has no pressure
   C. Pressure acts only on the wall
   D. The trough forces are vertical only
   Answer: A. Equal and opposite horizontal forces on the left and right quarters cancel. (Key point: Horizontal component)

2. What is F_H?
   A. h = 6 m: 2207 kN
   B. γ h^2/2 L with h = 4 m: 981 kN
   C. h = 2 m: 245 kN
   D. h = 4 m with γ = 9.81: 785 kN
   Answer: B. 12.26 x 16/2 x 10 = 981 kN. (Key point: Value of F_H)

3. What volume (area per metre) lies above the surface?
   A. Only the half-disc: 6.28 m^2
   B. 4 x 6 = 24 m^2
   C. A 4 x 4 rectangle plus a half-disc of radius 2 m: 22.28 m^2
   D. 16 + π = 19.1 m^2
   Answer: C. F_V = γ A L = 12.26 x 22.28 x 10 = 2732.5 kN downward. (Key point: Vertical component)

4. What is the resultant force?
   A. √(981^2 + 2732.5^2) ≈ 2903 kN, at θ = tan^-1(F_V/F_H) below horizontal
   B. 981 + 2732.5 = 3713 kN
   C. 2732.5 - 981 = 1752 kN
   D. 2732.5 kN
   Answer: A. Components are perpendicular, so combine them by Pythagoras. (Key point: Resultant)

## Related in Fluid Mechanics

- [Resultant force on curved surface ABCDE (S = 1.1)](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acurved-surface-abcde-s11-20m) — Fluid Mechanics, chapter 3
- [Resultant force on curved surface BCDEF](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Acurved-surface-bcdef-force) — Fluid Mechanics, chapter 3