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 need to use the concept of hydrostatic pressure and the center of pressure. Here are the steps to solve this problem:

Step 1: Find the radius of the circular disc:
Since the diameter is given as 125 cm, the radius will be half of the diameter. So, the radius (r) = 125 cm / 2 = 62.5 cm.

Step 2: Find the range of the vertical distance of the perimeter below the water surface:
The problem states that the distance of the perimeter measured vertically below the water surface varies between 60 cm and 150 cm. This means the disc can be at different angles with the water surface, causing the variation in the distance. We will consider the extreme cases here.

For the smallest distance (60 cm), the disc is completely submerged, and the center of pressure will be at the center of the disc.
For the largest distance (150 cm), the disc is almost parallel to the water surface, and the center of pressure will be at the midpoint between the center and the edge of the disc.
We will use these extreme cases to find the total force and the vertical distance of the center of pressure.

Step 3: Calculate the total force due to the water:
The total force on one side of the lamina is equal to the pressure at the center of pressure multiplied by the area of the disc.
The pressure at any point in a fluid is given by the formula 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 from the surface of the fluid.

When the disc is completely submerged (60 cm depth), the pressure at the center of pressure is P1 = ρg(60 cm). The area of the disc is given by A = πr^2.
So, the total force at this depth is F1 = P1 * A.

When the disc is almost parallel to the water surface (150 cm depth), the pressure at the center of pressure is P2 = ρg(150 cm). The area of the disc is still given by A = πr^2.
So, the total force at this depth is F2 = P2 * A.

Step 4: Find the vertical distance of the center of pressure below the surface:
Since the disc is symmetrical, the vertical distance of the center of pressure will be at the same position as the center of the disc. The center of the disc is at the midpoint of its diameter, which is half of the radius. So, the vertical distance of the center of pressure below the surface is equal to half the radius, or (1/2) * r.

Step 5: Plug in the values and calculate:
Substitute the known values into the above formulas and calculate the total force and the vertical distance of the center of pressure.

I hope this step-by-step explanation helps you solve the problem.