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 fluid pressure and the principles of buoyancy. Here's a step-by-step guide on how to solve it:
Step 1: Determine the vertical distance of the water surface below the center of the disc.
Given: The distance of the perimeter below the water surface varies between 60 cm and 150 cm.
Since the disc is symmetrical, its center coincides with the center of the circle. Therefore, the vertical distance of the water surface below the center of the disc is half of the distance of the perimeter measured vertically. Hence, the vertical distance ranges from 30 cm (60 cm / 2) to 75 cm (150 cm / 2).
Step 2: Calculate the area of the circular disc.
The area of a disc can be found using the formula: A = π * r^2, where r is the radius of the disc.
Given: The diameter of the disc is 125 cm.
Therefore, the radius (r) is half of the diameter, so r = 125 cm / 2 = 62.5 cm.
Now we can substitute the value of r into the formula to find the area (A) of the disc: A = π * 62.5^2.
Step 3: Calculate the total force due to the water acting on one side of the disc.
The force exerted by the water on the disc is equal to the pressure multiplied by the area.
The pressure varies linearly with depth according to Pascal's law.
Given: The pressure at the bottom of the disc is the maximum (150 cm) and at the top is the minimum (60 cm).
We need to find the average pressure exerted by the water.
Using the formula for average pressure, P_avg = (P1 + P2) / 2, where P1 and P2 are the pressures at the bottom and top respectively.
To convert the pressures from cm to meters, divide by 100 since 1 cm = 0.01 m.
Once we have the average pressure P_avg, we can calculate the force using the equation F = P_avg * A.
Step 4: Calculate the vertical distance of the center of pressure below the water surface.
To find the vertical distance of the center of pressure below the water surface, we need to consider the force acting on the disc and its moment about a reference point.
In this case, we can take the reference point as the center of the disc since it is at the same level as the center of pressure.
The moment of the force about the reference point is given by torque = force * distance.
We need to calculate the moment due to the total force acting on the disc as well as the force acting at the center of the disc.
Since both forces act vertically downwards, their moments will have opposite signs.
Equating the moments and dividing by the total force will give us the vertical distance of the center of pressure below the reference point.
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 water surface.