To what height should a reactangular cylinder having square base of length 10 cm be filled so that the total force on the bottom is equal to that on the sides

Options

A. 5 cm
B. 10 cm
C. 20 cm
D. 6.67 cm

Answer

To solve the problem, we need to determine the height \( h \) to which a rectangular cylinder with a square base of length 10 cm must be filled so that the total force on the bottom equals the total force on the sides.

### Step-by-Step Solution:

1. **Identify the dimensions of the cylinder:**
- The base of the cylinder is a square with a side length of \( L = 10 \) cm.
- The height of the liquid column is \( h \) cm.

2. **Calculate the area of the base:**
- The area \( A \) of the square base is given by:
\[ A = L^2 = (10 \, \text{cm})^2 = 100 \, \text{cm}^2 \]

3. **Calculate the force on the bottom of the cylinder:**
- The force \( F_b \) on the bottom due to the liquid is given by the pressure at the bottom multiplied by the area of the base. The pressure at the bottom due to a liquid column is given by:
\[ P = \rho g h \]
- Therefore, the force on the bottom is:
\[ F_b = P \cdot A = \rho g h \cdot A = \rho g h \cdot 100 \, \text{cm}^2 \]

4. **Calculate the force on the sides of the cylinder:**
- The force \( F_s \) on the sides can be calculated by integrating the pressure over the height of the liquid. The pressure at a height \( y \) is \( P(y) = \rho g y \).
- The differential force \( dF \) on a small strip of height \( dy \) and width \( L \) is:
\[ dF = P(y) \cdot L \cdot dy = \rho g y \cdot 10 \, dy \]
- To find the total force on the sides, integrate from \( 0 \) to \( h \):
\[ F_s = \int_0^h \rho g y \cdot 10 \, dy = \rho g \cdot 10 \cdot \int_0^h y \, dy \]
- The integral \( \int_0^h y \, dy = \frac{h^2}{2} \), so:
\[ F_s = \rho g \cdot 10 \cdot \frac{h^2}{2} = 5 \rho g h^2 \]

5. **Set the forces equal to each other:**
- We want the force on the bottom to equal the force on the sides:
\[ F_b = F_s \]
\[ \rho g h \cdot 100 = 5 \rho g h^2 \]

6. **Simplify the equation:**
- Cancel \( \rho g \) from both sides (assuming \( h \neq 0 \)):
\[ 100h = 5h^2 \]
- Rearranging gives:
\[ 5h^2 - 100h = 0 \]
- Factoring out \( h \):
\[ h(5h - 100) = 0 \]

7. **Solve for \( h \):**
- This gives us two solutions:
\[ h = 0 \quad \text{or} \quad 5h - 100 = 0 \]
- Solving \( 5h - 100 = 0 \):
\[ 5h = 100 \implies h = 20 \, \text{cm} \]

### Final Answer:
The height to which the cylinder should be filled is \( h = 20 \, \text{cm} \).