Project ForgeA hinged gate of length 5 m, inclined at 30° with the horizontal and with water mass on its left, is hinged at the top of the water surface. Density of water is 1000 kg/m³. The minimum mass of the gate in kg per unit width (perpendicular to the plane of paper), required to keep it closed is
- A5000
- B6600
- C7546
- 9623
Explanation
Vertical depth at the lower end = L·sin30°=2.5m. Centroid depth hc=(0+2.5)/2=1.25m. Hydrostatic force (per unit width) F=ρg·hc·L=1000×9.81×1.25×5=61312.5 N, acting at the center of pressure a distance scp=sc+Ic/(scA) along the gate from the hinge, where sc=L/2=2.5m and Ic=L³/12=125/12=10.417 m⁴ (unit width), A=L=5m²: scp=2.5+10.417/(2.5×5)=2.5+0.833=3.333m. Moment of water force about the hinge = F·scp=61312.5×3.333≈204,375 N·m. This must be balanced by the gate's own weight acting at its centroid, a horizontal distance (L/2)cos30°=2.165m from the hinge: mg×2.165=204,375, giving m=204,375/(9.81×2.165)≈9623 kg.
Source: GATE ME 2013 (Booklet Code A) (2013)
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