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 ?
Sure! Let's break down the problem into steps:
Step 1: Calculate the area of the circular disc.
- The area of a circle is given by the formula A = π * r^2, where r is the radius of the circle.
- Given the diameter of the circle is 125 cm, we can calculate the radius using the formula: radius = diameter/2.
- Calculate the radius: radius = 125 cm / 2 = 62.5 cm.
- Calculate the area: A = π * (62.5 cm)^2.
Step 2: Calculate the difference in water levels.
- The difference in water levels is given as the difference between the distance of the perimeter of the disc below the water surface (60 cm) and the maximum distance (150 cm).
- Calculate the difference: d = 150 cm - 60 cm.
Step 3: Calculate the total force due to the water.
- The force due to the water is equal to the pressure at each point multiplied by the area of the disc.
- The pressure at each point is given by the formula P = ρ * g * h, where ρ is the density of water, g is the acceleration due to gravity, and h is the vertical distance from the water surface to the point.
- Assuming the density of water is 1000 kg/m^3 and the acceleration due to gravity is 9.8 m/s^2, convert the lengths from cm to m: 1 cm = 0.01 m.
- Calculate the pressure: P = 1000 kg/m^3 * 9.8 m/s^2 * (d * 0.01 m).
Step 4: Calculate the total force.
- The total force is equal to the pressure multiplied by the area of the disc.
- Calculate the total force: F = P * A.
Step 5: Calculate the vertical distance of the center of pressure.
- The center of pressure is located at 2/3 of the vertical distance from the water surface to the bottom of the disc.
- Calculate the vertical distance of the center of pressure: hcp = (2/3) * d * 0.01 m.
Step 6: Convert the results back to cm.
- Convert the force and center of pressure back to cm if necessary.
I hope this helps! Let me know if you need any further clarification.
To solve this problem, we can use the concept of pressure and Archimedes' principle. Let's break down the solution into steps:
Step 1: Find the radius of the circular disc.
Since the diameter is given as 125 cm, the radius (r) can be calculated by dividing the diameter by 2.
r = 125 cm / 2 = 62.5 cm.
Step 2: Calculate the minimum and maximum lengths submerged in water.
The minimum length submerged is given as 60 cm, which means the disc is partially submerged.
The maximum length submerged is given as 150 cm, which means the disc is fully submerged.
Step 3: Calculating the area submerged.
The area submerged can be calculated using the formula for the area of a segment of a circle. Since the disc is partially submerged, we need to subtract the area of the triangle formed by the portion of the disc outside the water from the area of the sector of the circle submerged.
The area of the sector can be found using the formula: (θ/360) * π * r^2, where θ is the angle between the two radii forming the sector.
To find θ, we need to calculate the vertical distance between the minimum and maximum lengths submerged.
θ = 360° - (2 * sin^(-1)((r - h)/r))
where h is the vertical distance between the minimum and maximum lengths submerged.
The area of the triangle can be found using the formula: (1/2) * base * height, where the base is the length of the sector submerged and the height is the perpendicular distance between the two lengths submerged.
Finally, the area submerged is given by: Area_submerged = Area_sector - Area_triangle.
Step 4: Calculate the force due to water acting on the submerged side.
The force can be calculated using the formula: Force = Pressure * Area_submerged.
The pressure is given by the formula: Pressure = ρ * g * h, where ρ is the density of water and g is the acceleration due to gravity.
Step 5: Calculate the vertical distance of the center of pressure below the water surface.
The center of pressure is defined as the point through which the entire force due to the water can be considered to act. In a uniformly submerged lamina, the center of pressure lies at the centroid of the submerged area.
The vertical distance of the center of pressure below the water surface can be calculated using the formula:
Vertical distance of center of pressure = (2 * Depth of CB) / 3,
where Depth of CB is the distance from the centroid of the submerged area to the water surface.
By following these steps, you should be able to calculate 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.