A homogeneous rectangular gate is $4$ ft wide and $8$ ft long, and it weighs $800$ lb. The gate is hinged along its top edge and is submerged vertically in water so that its top edge is at the water surface. What is the minimum depth of water above the bottom edge of the gate required for the hydrostatic force to just begin to open the gate?

• A. $4$ ft
• B. $10$ ft
• C. $5$ ft
• D. $8$ ft

Answer: (B)

Verified step by step guidance

1. Identify the forces acting on the gate: the hydrostatic force due to water pressure pushing on the gate and the weight of the gate acting downward at its center of gravity.
2. Calculate the hydrostatic force on the gate using the formula for pressure on a submerged vertical surface: \(F = \rho g A \bar{h}\), where \(\rho\) is the density of water, \(g\) is acceleration due to gravity, \(A\) is the area of the gate, and \(\bar{h}\) is the depth of the centroid of the gate below the water surface.
3. Determine the location of the centroid of the gate, which is at half the height of the gate (since it is homogeneous and rectangular), so \(\bar{h} = \frac{\text{depth of water}}{2}\) if the top edge is at the water surface.
4. Calculate the moment caused by the hydrostatic force about the hinge at the top edge. The line of action of the hydrostatic force acts at the center of pressure, which is located below the centroid by a distance \(\frac{I_G}{A \bar{h}}\), where \(I_G\) is the second moment of area of the gate about its centroid.
5. Set the moment due to the hydrostatic force equal to the moment due to the weight of the gate about the hinge to find the minimum water depth that will just begin to open the gate. Solve the resulting equation for the water depth.