A 2 m × 3 m rectangular gate is vertical with its top edge at a depth of 1 m. What is the total hydrostatic force on the gate? (ρ = 1000 kg/m³, g = 9.81 m/s²)
Think about your answer, then reveal below.
Model answer: The centroid depth h_c = 1 + 1.5 = 2.5 m. F = ρg·h_c·A = 1000 × 9.81 × 2.5 × 6 = 147,150 N ≈ 147 kN.
The centroid of the gate is at the midpoint of its 3 m height, so 1.5 m below the top edge, giving h_c = 2.5 m. Multiplying centroidal pressure by area gives the resultant. Note that A = 2 × 3 = 6 m².
Question 2 Short Answer
Why does the center of pressure always lie below the centroid for a submerged surface?
Think about your answer, then reveal below.
Model answer: Because hydrostatic pressure increases linearly with depth, the lower portion of the surface experiences greater pressure than the upper portion. The resultant force is therefore weighted toward the bottom of the surface, placing the center of pressure below the geometric centroid.
Only if pressure were uniform (infinite depth approximation) would the two points coincide. The offset I_c/(y_c·A) quantifies this shift; it decreases as depth increases because at great depth the pressure variation across the surface becomes a smaller fraction of the total pressure.