Problem Statement
A square aperture in the vertical side of a tank has one diagonal vertical and is completely covered by a plane plate hinged along one of the upper sides of the aperture. The diagonals of the aperture are 2 m long and the tank contains a liquid of specific gravity 1.15. The centre of the aperture is 1.5 m below the free surface. Calculate the thrust exerted on the plate by the liquid and the position of its centre of pressure.
Given Data
- Diagonals of aperture, \( d = 2 \, \text{m}\)
- Specific gravity of liquid, \( S = 1.15 \)
- Density of liquid, \( \rho = 1.15 \times 1000 = 1150 \, \text{kg/m}^3 \)
- Depth of C.G. from free surface, \( \bar{h} = 1.5 \, \text{m}\)
Diagram
The square aperture oriented with one diagonal vertical.
Solution
(i) Total Thrust (Force)
The area of the square aperture can be calculated from its diagonals. A square with diagonal \(d\) is composed of two triangles, each with a base of \(d\) and a height of \(d/2\).
The total thrust (force) on the plate is given by:
(ii) Centre of Pressure (\(h^*\))
The centre of pressure is found using \( h^* = \frac{I_G}{A\bar{h}} + \bar{h} \). First, we need the moment of inertia (\(I_G\)) about the centroidal axis parallel to the free surface. For a square oriented with diagonals vertical and horizontal, the horizontal diagonal passes through the centroid.
The M.O.I. of the square about its horizontal diagonal is the sum of the M.O.I. of the two triangles (top and bottom) about their common base.
Now, calculate the centre of pressure:
Total Thrust (Force): \( F \approx 33845 \, \text{N} \)
Centre of Pressure: \( h^* \approx 1.611 \, \text{m} \) below the free surface
Explanation of Concepts
Thrust (Hydrostatic Force): This is the total force exerted by the liquid on the plate. It's calculated by multiplying the pressure at the plate's geometric center (centroid) by the plate's area. This gives an average force value over the entire surface.
Centre of Pressure: This is the specific point on the plate where the total thrust can be considered to act. Since pressure increases with depth, this point is always vertically below the centroid of the plate. Calculating its exact location is crucial for understanding the rotational forces (moments) on the plate, especially important here since the plate is hinged.
Physical Meaning
The plate must be strong enough to withstand a force of nearly 34,000 N, which is roughly equivalent to the weight of 3.4 metric tons.
The force acts at a depth of 1.611 m, which is about 11 cm below the center of the aperture (1.5 m). Since the plate is hinged at the top, this downward shift in the force's point of application creates a torque that will try to swing the bottom of the plate open. The hinge and any locking mechanism must be designed to resist this specific turning moment.
