Project ForgeA vertical hinged gate holds back two stacked fluid layers: a 1m-deep top layer of density 1000 kg/m³ and a 2m-deep bottom layer of density 2000 kg/m³, with the hinge at the base. Atmospheric pressure (100 kPa) acts on both the free surface and the back of the gate, and can be ignored via gauge pressure. The moment per unit width about the base hinge is ______ kNm/m. (Rounded to one decimal place, g=9.8 m/s²)
Explanation
Measuring depth y from the free surface: for 0≤y≤1, P(y)=1000×9.8×y. For 1≤y≤3, P(y)=9800+2000×9.8×(y−1). Moment about the hinge (at y=3) per unit width: M=∫₀³P(y)(3−y)dy, split into the two layers. The top-layer contribution is ≈11,433 N·m/m and the bottom-layer contribution is ≈45,733 N·m/m, for a total of ≈57,167 N·m/m ≈57.2 kNm/m.
Source: GATE ME 2023 Official Paper (2023)
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