Problem Statement
Determine the total pressure and depth of centre of pressure on a plane rectangular surface of 1 m wide and 3 m deep when its upper edge is horizontal and (a) coincides with water surface (b) 2 m below the free water surface.
Given Data
- Width of surface, \( b = 1 \, \text{m} \)
- Depth of surface, \( d = 3 \, \text{m} \)
- Density of Water, \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to Gravity, \( g = 9.81 \, \text{m/s}^2 \)
Solution
(a) Upper Edge Coincides with Water Surface
Total Pressure (Force), \(F\)
The total pressure is the force acting on the surface, given by \( F = \rho g A \bar{h} \).
Depth of Centre of Pressure, \(h^*\)
The depth of the centre of pressure is given by \( h^* = \frac{I_G}{A\bar{h}} + \bar{h} \).
(b) Upper Edge 2 m Below Water Surface
Total Pressure (Force), \(F\)
The area \(A\) remains the same, but the depth to the centroid \(\bar{h}\) changes.
Depth of Centre of Pressure, \(h^*\)
The moment of inertia \(I_G\) is the same, but we use the new value for \(\bar{h}\).
Case (a): Total Pressure \( F = 44145 \, \text{N} \), Centre of Pressure \( h^* = 2 \, \text{m} \).
Case (b): Total Pressure \( F = 103005 \, \text{N} \), Centre of Pressure \( h^* \approx 3.714 \, \text{m} \).
Explanation of Concepts
Total Pressure (Hydrostatic Force): This is the net force exerted by a fluid at rest on a submerged surface. It is calculated by multiplying the pressure at the centroid (center of area) of the surface by the total area of the surface. The pressure at the centroid represents the average pressure acting on the surface.
Centre of Pressure: This is the specific point on the submerged surface where the total hydrostatic force can be considered to act. Because fluid pressure increases with depth, the pressure on the lower portion of a surface is greater than on the upper portion. This causes the centre of pressure to always be located below the centroid of the surface (for non-horizontal surfaces).
