A vertical gate closes a horizontal tunnel 4 m high and 3 m wide running full with water. The bottom of the vertical gate is located at a distance of 8 m from the free water surface. Determine the total fluid force (in kN) acting on the gate.
Try it first: pick an option, then read the explanation.
Asked in SSC JE 10/10/2023 (05pm-07pm)
Detailed Explanation
For a vertical rectangular gate submerged below the free surface, the total fluid thrust is determined by the depth to the geometric centroid of the gate area.
1. Establish the geometry from the problem description:
- Height of gate \(H = 4\text{ m}\)
- Width of gate \(W = 3\text{ m}\)
- Depth to bottom of gate \(h_{\text{bottom}} = 8\text{ m}\)
- Depth to top of gate \(h_{\text{top}} = 8 - 4 = 4\text{ m}\)
2. Calculate the depth of the centroid (\(\bar{h}\)):
\[ \bar{h} = h_{\text{top}} + \frac{H}{2} = 4 + \frac{4}{2} = 6\text{ m} \]
3. Calculate the surface area of the gate:
\[ A = H \times W = 4\text{ m} \times 3\text{ m} = 12\text{ m}^2 \]
4. Calculate the total hydrostatic thrust \(P\):
\[ P = \rho g A \bar{h} = 1000\text{ kg/m}^3 \times 9.81\text{ m/s}^2 \times 12\text{ m}^2 \times 6\text{ m} \]
\[ P = 706,320\text{ N} = 706.32\text{ kN} \approx 706\text{ kN} \]
Why “706” Is Correct
With the gate extending from \(4\text{ m}\) to \(8\text{ m}\) below the water surface, its centroid lies midway at \(\bar{h} = 6\text{ m}\). The product \(\gamma A \bar{h} = 9.81 \times 12 \times 6 = 706.32\text{ kN}\), which rounds to \(706\text{ kN}\).
Why the Other Options Are Wrong
- A. 720: 720 kN results from using g = 10 m/s²: 1000 × 10 × 12 × 6 = 720 kN.
- C. 740: 740 kN results from mistakenly using the bottom depth of 8 m as the centroid depth.
- D. 730: 730 kN is an intermediate arithmetic error.
Key Point
Submerged gate thrust P = γ · A · h̄; always calculate centroid depth h̄ from the free surface.
Key Concepts for Students
- The top of the gate is at h_bottom − H = 8 − 4 = 4 m.
- Centroid depth h̄ = (h_top + h_bottom) / 2 = (4 + 8) / 2 = 6 m.
- If g = 10 m/s² had been specified, P would be 720 kN; using standard g = 9.81 m/s² gives 706.32 kN.
Practice more: this is question 6 in Hydrostatic Pressure MCQs – Part 1. Try the whole set and check every answer instantly.