to what height should a cylindrical vessel of diameter d be filled with a liquid so that due to force on the verticle surface of the vessel be equal to the force on the bottom
What he has did . i can't understand
Really this soln isn't apt.
To find the height to which a cylindrical vessel should be filled with a liquid so that the force on the vertical surface is equal to the force on the bottom, you need to understand the concept of pressure and the basic principles of fluid mechanics.
The force on the vertical surface of the cylindrical vessel is due to the pressure exerted by the liquid. The pressure at any point in a fluid is defined as the force per unit area exerted by the fluid on its surroundings. In this case, the force on the vertical surface is equal to the pressure exerted by the liquid multiplied by the surface area of the vertical side of the vessel.
Similarly, the force on the bottom of the vessel is also due to the pressure exerted by the liquid. However, in this case, the pressure is acting on the entire base area of the vessel.
Now, let's break down the steps to find the height to which the vessel should be filled:
1. Identify the forces:
- Force on the vertical surface: This is the force due to the pressure exerted by the liquid on the vertical side of the cylinder. It is equal to the pressure multiplied by the surface area of the vertical side of the vessel.
- Force on the bottom: This is the force due to the pressure exerted by the liquid on the bottom of the vessel. It is equal to the pressure multiplied by the entire base area of the vessel.
2. Equate the forces:
Since we want the forces on the vertical surface and the bottom to be equal, we set up an equation:
Force on the vertical surface = Force on the bottom
3. Calculate the pressures:
- Pressure on the vertical surface: Divide the force on the vertical surface by the surface area of the vertical side of the vessel.
- Pressure on the bottom: Divide the force on the bottom by the entire base area of the vessel.
4. Set the pressures equal to each other in the equation:
Pressure on the vertical surface = Pressure on the bottom
5. Substitute the formulas for pressure:
(Pressure is defined as force divided by area)
[Force on vertical surface / (Surface area of vertical side)] = [Force on bottom / (Entire base area)]
6. Simplify the equation:
The terms involving force cancel out, leaving us with an equation relating the surface area of the vertical side, the entire base area, and the desired height of liquid.
7. Solve for height:
Rearrange the equation to solve for the height of the liquid in terms of the given diameter and other known quantities.
By following these steps, you should be able to find the height to which the cylindrical vessel should be filled with liquid so that the forces on the vertical surface and bottom are equal.
force on side walls = p halfway up * area of walls
= (rho g h/2) pi d h
force on bottom = rho g h (pi d^2/4)
so
2 rho g h^2 pi d = rho g h pi d^2
2 h = d
h = d/2