To what height should a cyclindrical vessel be filled with a homogeneous liquid to make the force with which the liquid pressure on the sides of the vessel equal to the force exerted by the liquid on the bottom of the vessel ?

Options

A. equal to the radius
B. less than radius
C. more than radius
D. four times of radius

Explanation

To solve the problem of determining the height to which a cylindrical vessel should be filled with a homogeneous liquid so that the force exerted by the liquid on the sides of the vessel equals the force exerted by the liquid on the bottom of the vessel, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( H \) be the height of the liquid in the cylinder.
- Let \( r \) be the radius of the cylinder.
- Let \( \rho \) be the density of the liquid.
- Let \( g \) be the acceleration due to gravity.

2. **Calculate the Weight of the Liquid**:
- The volume of the liquid in the cylindrical vessel is given by \( V = \pi r^2 H \).
- The weight of the liquid, \( W \), can be calculated using the formula:
\[ W = \text{Volume} \times \text{Density} \times g = V \cdot \rho \cdot g = \pi r^2 H \rho g \]
- Name this equation as **Equation 1**:
\[ W = \pi r^2 H \rho g \]

3. **Calculate the Force Exerted by the Liquid on the Bottom**:
- The pressure at the bottom of the vessel due to the liquid column is given by:
\[ P = \rho g H \]
- The area of the bottom of the cylinder is \( A = \pi r^2 \).
- Therefore, the force \( F_b \) exerted by the liquid on the bottom is:
\[ F_b = P \cdot A = (\rho g H) \cdot (\pi r^2) = \pi r^2 \rho g H \]

4. **Calculate the Force Exerted by the Liquid on the Sides**:
- The mean pressure on the sides of the vessel is given by the average pressure, which is \( \frac{\rho g H}{2} \) (since pressure varies linearly from 0 at the top to \( \rho g H \) at the bottom).
- The total force \( F_s \) exerted by the liquid on the sides of the vessel is:
\[ F_s = \text{Mean Pressure} \times \text{Area of the sides} = \left(\frac{\rho g H}{2}\right) \cdot (2\pi r H) = \pi r H \rho g H \]
- Name this equation as **Equation 2**:
\[ F_s = \pi r H^2 \rho g \]

5. **Set the Forces Equal**:
- To find the height \( H \) such that the force on the sides equals the force on the bottom, we set \( F_s = W \):
\[ \pi r H^2 \rho g = \pi r^2 H \rho g \]

6. **Simplify the Equation**:
- Cancel out common terms (\( \pi \), \( \rho \), \( g \), and one \( r \)):
\[ H^2 = r H \]

7. **Solve for Height \( H \)**:
- Rearranging gives:
\[ H^2 - rH = 0 \]
- Factoring out \( H \):
\[ H(H - r) = 0 \]
- This gives us two solutions: \( H = 0 \) or \( H = r \). Since \( H = 0 \) is not meaningful in this context, we have:
\[ H = r \]

### Conclusion:
The height to which the cylindrical vessel should be filled with the liquid is equal to the radius of the cylinder:
\[ H = r \]