A closed tank, rectangular in plan with vertical sides, is 1.8m deep and contains water to a depth of 1.2m. The enclosed space above the water is pressurized with air up to 35kPa. If the length of one wall of the tank is 3m, determine the resultant force on this wall and the height of the centre of pressure above the base.
To determine the resultant force on the wall and the height of the center of pressure above the base, we need to consider the pressure distribution acting on the wall.
The pressure acting on the wall can be divided into two components: the hydrostatic pressure due to the water and the pressure due to the air above the water.
1. Hydrostatic Pressure due to the Water:
The hydrostatic pressure increases linearly with depth. The pressure at any depth is given by the equation:
P_water = ρ * g * h
Where:
- P_water is the pressure exerted by the water
- ρ is the density of water (assumed to be 1000 kg/m³)
- g is the acceleration due to gravity (assumed to be 9.81 m/s²)
- h is the depth from the surface of the water (1.2m in this case)
So, the pressure exerted by the water can be calculated as:
P_water = 1000 * 9.81 * 1.2
2. Pressure due to the Air:
The pressure due to the air is constant and given as 35kPa.
Now, let's calculate the resultant force acting on the wall and the height of the center of pressure above the base.
Resultant Force:
The resultant force can be obtained by summing up the pressure forces acting on the wall. Since the pressure forces act normal (perpendicular) to the wall, we do not need to consider the horizontal components.
Resultant Force = (P_water + P_air) * Area_of_the_Wall
Area_of_the_Wall can be calculated as the product of the length of the wall and the depth of the tank:
Area_of_the_Wall = 3 * 1.8
So, the resultant force can be calculated as:
Resultant Force = (P_water + P_air) * (3 * 1.8)
Height of the Center of Pressure:
The height of the center of pressure can be calculated using the equation:
Height_of_Center_of_Pressure = (Moment_due_to_Pressure_forces) / (Resultant Force)
Moment_due_to_Pressure_forces = (P_water * depth_of_the_Water * Area_of_the_Wall) + (P_air * (height_of_the_air_above_water - depth_of_the_Water) * Area_of_the_Wall)
So, the height of the center of pressure can be calculated as:
Height_of_Center_of_Pressure = (Moment_due_to_Pressure_forces) / (Resultant Force)
By plugging in the values, you can calculate the resultant force on the wall and the height of the center of pressure above the base.