Question

A square lamina (each side equal to 2m) is submerged vertically in water such that the upper edge of the lamina is at a depth of 0.5 m from the free surface. What will be the total water pressure (in kN) on the lamina? (take \( \gamma_w = 10 \, \text{kN/m}^3 \))

• A. 20 kN
• B. 40 kN
• C. 60 kN
• D. 80 kN

Hint:

When calculating the total hydrostatic force on a submerged surface, use the pressure at the centroid of the surface multiplied by the area. For a vertical surface, the centroid depth accounts for the linear variation of pressure with depth, simplifying the calculation of the total force.

Answer 1

The Correct Option is C

Step 1: Identify the given data.
We are given a square lamina with each side equal to 2 m, submerged vertically in water. The upper edge of the lamina is at a depth of 0.5 m from the free surface. The unit weight of water is \( \gamma_w = 10 \, \text{kN/m}^3 \). We need to find the total water pressure (force) on the lamina in kN. The options are:
(1) 20 kN
(2) 40 kN
(3) 60 kN
(4) 80 kN
Step 2: Determine the position and dimensions of the lamina.
The lamina is a square with side length 2 m, so its area is:
\[ A = 2 \, \text{m} \times 2 \, \text{m} = 4 \, \text{m}^2 \] The lamina is submerged vertically, with its upper edge at a depth of 0.5 m from the free surface. Since the side length is 2 m, the depth of the lower edge is:
\[ 0.5 \, \text{m} + 2 \, \text{m} = 2.5 \, \text{m} \] Thus, the lamina extends from a depth of 0.5 m to 2.5 m below the free surface.
Step 3: Calculate the pressure and total force on the lamina.
The water pressure at a depth \( h \) below the free surface is given by:
\[ p = \gamma_w \cdot h \] Since the lamina is vertical, the pressure varies linearly from the top to the bottom. To find the total force, we need the pressure at the centroid of the lamina (the average pressure), then multiply by the area.
The centroid of the square lamina is at its geometric center. Since the lamina extends from 0.5 m to 2.5 m in depth, the depth to the centroid is:
\[ h_{\text{centroid}} = 0.5 \, \text{m} + \frac{2 \, \text{m}}{2} = 0.5 \, \text{m} + 1 \, \text{m} = 1.5 \, \text{m} \] The pressure at the centroid is:
\[ p_{\text{centroid}} = \gamma_w \cdot h_{\text{centroid}} = 10 \, \text{kN/m}^3 \times 1.5 \, \text{m} = 15 \, \text{kN/m}^2 \] The total force (or total water pressure in kN) on the lamina is the pressure at the centroid multiplied by the area:
\[ F = p_{\text{centroid}} \times A = 15 \, \text{kN/m}^2 \times 4 \, \text{m}^2 = 60 \, \text{kN} \]
Step 4: Verify the calculation.
Let’s confirm by considering the pressure distribution:
Pressure at the top edge (depth 0.5 m): \( 10 \times 0.5 = 5 \, \text{kN/m}^2 \)
Pressure at the bottom edge (depth 2.5 m): \( 10 \times 2.5 = 25 \, \text{kN/m}^2 \)
The average pressure is:
\[ \frac{5 + 25}{2} = 15 \, \text{kN/m}^2 \] This matches the pressure at the centroid, confirming our approach. The total force is indeed 60 kN.
Step 5: Select the correct option.
The total water pressure on the lamina is 60 kN, which matches option (3).
\[ \boxed{60 \, \text{kN}} \]