Problem Statement
Determine the total pressure on a circular plate of diameter 1.5 m which is placed vertically in water in such a way that the centre of the plate is 2 m below the free surface of water. Find the position of the centre of pressure also.
Given Data
- Diameter of plate, \( D = 1.5 \, \text{m} \)
- Depth of centre of plate, \( \bar{h} = 2 \, \text{m} \)
- Density of Water, \( \rho = 1000 \, \text{kg/m}^3 \)
- Acceleration due to Gravity, \( g = 9.81 \, \text{m/s}^2 \)
Solution
Total Pressure (Force), \(F\)
The total pressure is the force acting on the plate, given by the formula \( F = \rho g A \bar{h} \).
Depth of Centre of Pressure, \(h^*\)
The depth of the centre of pressure for a vertically submerged surface is given by \( h^* = \frac{I_G}{A\bar{h}} + \bar{h} \).
Total Pressure (Force): \( F \approx 34668.5 \, \text{N} \) or \( 34.67 \, \text{kN} \).
Position of Centre of Pressure: \( h^* \approx 2.07 \, \text{m} \) below the free surface.
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).
