A circular disc having a diameter of 125cm is immersed in water so that the distance of its perimeter measured vertically below the water surface varies between 60cm and 150cm. Find the total force due to the water acting on one side of the lamina, and the vertical distance of the centre of pressure below the surface.
Can someone please show me how to solve this problem? With clear step ?
Instead of doing the integral of pressure over the lower surface, try Archimedes.
Imagine a vertical shaft, elliptical in cross section, in the water ending at your angled disc.
The total force up on it will be the buoyancy or weight (mg) of water displaced.
U dint solve the question, that's is no mass
Sure! I can help you solve this problem step-by-step. Let's start with finding the total force due to water acting on one side of the lamina.
Step 1: Find the radius of the circular disc.
The diameter is given as 125 cm, so the radius (r) can be calculated by dividing the diameter by 2:
r = 125 cm / 2 = 62.5 cm
Step 2: Calculate the area of the circular disc.
The area of a circle is given by the formula A = πr^2, where π is a constant value (approximately 3.14159).
A = π(62.5 cm)^2
A ≈ 3.14159 * (62.5^2) cm^2
A ≈ 3.14159 * 3906.25 cm^2
A ≈ 12252.39 cm^2
Step 3: Determine the depth of the disc in water.
The difference in perimeter measured vertically below the water surface will give us the depth.
The range of the vertical perimeter is given as 60 cm to 150 cm. The maximum and minimum levels are reached when the disk is vertical and horizontal, respectively. So, the maximum depth of the disc is 125 cm (total diameter), and the minimum depth is 0 cm.
Step 4: Calculate the average depth of the disc.
To find the average depth, we add the maximum and minimum depths and divide by 2.
Average depth = (maximum depth + minimum depth) / 2
Average depth = (125 cm + 0 cm) / 2
Average depth = 62.5 cm
Step 5: Calculate the centroid of the submerged area.
The centroid is the geometric center of the submerged area, which is also the center of pressure.
In a circular disc, the centroid is located at 2/3 of the total depth from the water surface.
Step 6: Calculate the distance of the center of pressure below the water surface.
The distance of the center of pressure below the water surface can be found by multiplying the average depth by 2/3.
Distance of center of pressure = Average depth * 2/3
Distance of center of pressure = 62.5 cm * (2/3)
Distance of center of pressure = 41.67 cm
Step 7: Calculate the force due to water acting on one side of the lamina.
The force due to water can be found using the equation:
Force = Pressure x Area
where Pressure is the pressure due to water and Area is the area of the circular disc.
Step 8: Calculate the pressure due to water.
The pressure due to water can be calculated using the formula:
Pressure = density x gravity x depth
where density is the density of water (usually taken as 1000 kg/m^3) and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).
We need to convert the average depth from cm to meters before calculation.
Average depth = 62.5 cm / 100 = 0.625 m
Pressure = 1000 kg/m^3 * 9.8 m/s^2 * 0.625 m
Pressure = 6137.5 N/m^2 (or Pascals)
Step 9: Calculate the force due to water acting on one side of the lamina.
Force = Pressure x Area
Force = 6137.5 N/m^2 * 0.12252 m^2 (convert area from cm^2 to m^2)
Force ≈ 751.34 N (approximately)
Therefore, the total force due to the water acting on one side of the lamina is approximately 751.34 N. The vertical distance of the center of pressure below the surface is approximately 41.67 cm.
Sure! To solve this problem, we need to use the concept of hydrostatic pressure and the principle of moments.
Step 1: Determine the area of the circular disc.
The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius. In this case, the diameter is given as 125 cm, so the radius (r) is half of the diameter, which is 62.5 cm. Substitute this value into the formula to find the area of the circular disc.
Step 2: Determine the range of the vertical distance of the perimeter below the water surface.
We are given that the range of the distance of the perimeter below the water surface varies between 60 cm and 150 cm.
Step 3: Calculate the pressure at the top and bottom of the disc.
Using the concept of hydrostatic pressure, we know that the pressure at a given depth in a fluid is given by P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth.
At the top of the disc, the depth is 60 cm or 0.6 m (converted from cm to meters). Calculate the pressure at the top using the given depth and the density of water (ρ = 1000 kg/m^3) and acceleration due to gravity (g = 9.8 m/s^2).
At the bottom of the disc, the depth is 150 cm or 1.5 m. Calculate the pressure at the bottom using the same formula.
Step 4: Calculate the average pressure acting on the disc.
The average pressure acting on the disc can be calculated by taking the average of the pressures at the top and bottom. Add the pressures at the top and bottom and divide by 2.
Step 5: Calculate the total force on the disc.
The total force acting on the disc can be calculated by multiplying the average pressure by the area of the disc. This gives us the force due to the water acting on one side of the lamina.
Step 6: Calculate the moment of the force about the center of the disc.
To find the vertical distance of the center of pressure below the surface, we need to calculate the moment of the force due to the water about the center of the disc. The moment is given by the formula M = Fd, where M is the moment, F is the force, and d is the perpendicular distance from the line of action of the force to the center of the disc.
In this case, the perpendicular distance is half the range of the vertical distance of the perimeter below the water surface.
Step 7: Calculate the vertical distance of the center of pressure below the surface.
Finally, divide the moment of the force by the total force to find the vertical distance of the center of pressure below the surface.
That's it! By following these steps, you should be able to solve the problem and find the total force due to the water acting on one side of the lamina and the vertical distance of the center of pressure below the surface.