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.
- A720
- B740
- C730
- D706Correct
Solution
Concept
Total hydrostatic force on a submerged plane surface = specific weight × area × depth of centroid.
Formula
F = ρ g A h̄
Step-by-step solution
1. Area of gate A = 4 × 3 = 12 m² 2. Bottom of gate is 8 m below free surface, gate height = 4 m, so top is at 8 - 4 = 4 m depth 3. Depth of centroid h̄ = 4 + 4/2 = 6 m 4. F = ρ g A h̄ = 1000 × 9.81 × 12 × 6 = 706320 N 5. F ≈ 706 kN
Explanation
The gate is a vertical rectangle fully under water. The force equals the water pressure at the centroid times the area. The centroid is 6 m deep, so the force is about 706 kN.
Why the other options are wrong
720 kN uses g = 10 m/s². 740 kN and 730 kN do not match any correct depth or g value.
Shortcut
F = 9.81 × 12 × 6 ≈ 706 kN directly in kN units.
Common mistake
Taking 8 m (the bottom depth) as centroid depth, or using g = 10 m/s² to get 720 kN.
Correct answer: D) 706