Problem Statement
A rectangular plane surface 3 m wide and 4 m deep lies in water in such a way that its plane makes an angle of 30° with the free surface of water. Determine the total pressure force and position of centre of pressure, when the upper edge is 2 m below the free surface.
Given Data
- Width of plane, \( b = 3 \, \text{m}\)
- Depth of plane, \( d = 4 \, \text{m}\)
- Angle with free surface, \( \theta = 30^\circ \)
- Depth of upper edge below free surface = \( 2 \, \text{m}\)
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \)
Diagram
Illustration of the inclined rectangular plane submerged in water.
Solution
(i) Total Pressure Force (\(F\))
The total pressure force (or hydrostatic thrust) is given by the formula:
First, we calculate the area of the plate, \( A \):
Next, we find the vertical depth of the centroid (C.G.) of the plate, \( \bar{h} \). The centroid is at a distance of \(d/2\) from the upper edge along the plate. The vertical depth is the sum of the upper edge's depth and the vertical component of the distance to the centroid.
Now, we can calculate the total force:
(ii) Centre of Pressure (\(h^*\))
The centre of pressure is the point where the total force acts. Its vertical depth, \(h^*\), is calculated using the parallel axis theorem for an inclined surface:
Where \(I_G\) is the moment of inertia of the rectangular area about its centroidal axis parallel to the free surface.
Now, substitute the values into the formula for \(h^*\):
Total Pressure Force: \( F \approx 353.16 \, \text{kN} \)
Position of Centre of Pressure: \( h^* \approx 3.111 \, \text{m} \) below the free surface
Explanation of Concepts
Total Pressure Force: This is the net force that the water exerts on the submerged rectangular plate. It's calculated based on the pressure at the plate's geometric center (centroid) and the plate's total area. This force acts perpendicular to the surface.
Centre of Pressure: This is the specific point on the plate where the total pressure force can be considered to act. Since water pressure increases with depth, the force is greater on the lower part of the plate than the upper part. This causes the centre of pressure to be located vertically below the centroid. Its precise location is critical for structural design, such as determining the forces on hinges or supports.
Physical Meaning
The total force of over 353 kN is substantial, equivalent to the weight of about 36 metric tons (or about 79,000 pounds). Any structure holding this plate must be designed to withstand this immense load.
The force acts at a depth of 3.111 m, which is 0.111 m (11.1 cm) vertically below the centroid's depth of 3 m. This offset creates a turning moment (torque) on the plate. If the plate were a gate hinged at its top edge, this force would try to push the bottom edge open. The hinge and locking mechanisms must be strong enough to resist this rotational effect.
