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 ?
To solve this problem, we will use the principles of fluid pressure and the concept of the center of pressure. Here are the step-by-step instructions to find the total force and the vertical distance of the center of pressure below the surface:
1. First, let's calculate the radius of the circular disc. The diameter is given as 125cm, so the radius is half of that: 125cm / 2 = 62.5cm.
2. The vertical distance measured below the water surface varies between 60cm and 150cm. Let's call this height h, where h ranges from 60cm to 150cm.
3. We know that the pressure exerted by the water depends on the depth from the surface. The formula to calculate the pressure is P = ρ * g * h, where P is the pressure, ρ is the density of the water (which we'll assume as a constant), g is the acceleration due to gravity, and h is the depth.
4. Now, let's find the pressure at the upper and lower edges of the disc. At the upper edge, the depth is (h - r), where r is the radius of the disc. At the lower edge, the depth is (h + r).
5. The total force exerted by the water can be found by integrating the pressure over the curved surface of the disc. The formula to calculate the force is F = ∫ P * dA, where F is the force, P is the pressure, and dA is an infinitesimally small area element on the curved surface.
6. For a circular disc, the pressure is constant at every point along a horizontal line on the surface. Therefore, we can calculate the force as F = p * A, where p is the pressure at the lower edge, and A is the total area of the disc.
7. The area of a circular disc is A = π * r^2, where r is the radius. So, the force becomes F = p * π * r^2.
8. To determine the vertical distance of the center of pressure below the surface, we need to consider the centroid of the circular disc. The center of pressure lies on the line passing through the centroid and is in equilibrium with the total force exerted by the water.
9. The centroid of a circle lies at a distance of (4r/3π) below the center of the circle. So, the vertical distance of the center of pressure below the surface is given by hcp = h - (4r/3π).
10. Now, substitute the values of r, p, and h in the above formulas to calculate the total force and the vertical distance of the center of pressure.
Following these steps, you should be able to 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.