A tank has height and width as 4 m and 3 m, respectively. Determine the total water force, in kN, acting on the bottom of the tank when it is completely filled with water. Take density of water as 1000 kg/m³ and acceleration due to gravity as 9.81 m/sec² (Take the length of tank as 3 m.)

Asked in SSC JE 07/06/2024 (09am-11am)

Detailed Explanation

The total hydrostatic force acting on a flat horizontal bottom of a liquid container is determined by the uniform hydrostatic pressure at that depth multiplied by the surface area of the bottom.

1. Identify the given parameters:

Tank height (depth of water) \(h = 4\text{ m}\)
Tank width \(w = 3\text{ m}\)
Tank length \(l = 3\text{ m}\)
Water density \(\rho = 1000\text{ kg/m}^3\)
Gravitational acceleration \(g = 9.81\text{ m/s}^2\)

2. Calculate the hydrostatic pressure intensity at the bottom:

\[ p = \rho g h = 1000 \times 9.81 \times 4 = 39,240\text{ N/m}^2 = 39.24\text{ kPa} \]

3. Calculate the bottom area of the tank:

\[ A = l \times w = 3\text{ m} \times 3\text{ m} = 9\text{ m}^2 \]

4. Calculate the total hydrostatic force acting on the bottom:

\[ F = p \times A = 39,240\text{ N/m}^2 \times 9\text{ m}^2 = 353,160\text{ N} = 353.16\text{ kN} \]

Why “353.16” Is Correct

Because the bottom of the tank is horizontal and situated at a constant depth of \(h = 4\text{ m}\), the hydrostatic pressure is uniform across its entire area. Multiplying this uniform bottom pressure of \(39.24\text{ kN/m}^2\) by the floor area of \(9\text{ m}^2\) yields exactly \(353.16\text{ kN}\).

Total hydrostatic force on the bottom of a rectangular water tank (4 m deep, 3 m × 3 m base).

Why the Other Options Are Wrong

B. 345.13: 345.13 kN results from an arithmetic error or mistakenly using g ≈ 9.58 m/s².
C. 351.26: 351.26 kN results from rounding intermediate pressure values or using g = 9.75 m/s².
D. 400.57: 400.57 kN results from incorrectly using g = 10 m/s² along with an inflated base area calculation.

Key Point

Total bottom force F = ρ·g·h·A; for a horizontal base, pressure is uniform at all points.

Key Concepts for Students

Hydrostatic pressure on any horizontal plane at depth h is uniform and given by p = ρgh.
Total hydrostatic thrust on a horizontal surface equals the weight of the liquid column vertically above it.
Tank wall orientation does not alter bottom pressure; pressure depends strictly on vertical liquid depth (hydrostatic paradox).

Practice more: this is question 1 in Hydrostatic Pressure MCQs – Part 1. Try the whole set and check every answer instantly.