Question

To what height should a rectangular 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 ?

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

Hint:

For vertical surfaces, use average pressure \(= \frac{\rho g h}{2}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The force on the bottom is due to uniform pressure at depth \( h \). The force on the side walls is due to an average pressure over the height \( h \).
: Key Formula or Approach:
1. Force on bottom \( F_b = P \times \text{Area}_{\text{bottom}} = (\rho gh) \times l^2 \).
2. Force on one side \( F_{s1} = P_{\text{avg}} \times \text{Area}_{\text{side}} = \left( \frac{\rho gh}{2} \right) \times (l \times h) \).
3. Total force on all 4 sides \( F_s = 4 \times F_{s1} \).
Step 2: Detailed Explanation:
Let \( l = 10 \text{ cm} \) and \( h \) be the height.
- Force on bottom: \( F_b = \rho gh l^2 \).
- Force on four sides: \( F_s = 4 \times \left( \frac{\rho gh}{2} \times l \times h \right) = 2\rho gl h^2 \).
According to the problem, \( F_b = F_s \):
\[ \rho gh l^2 = 2 \rho gl h^2 \]
Canceling common terms (\( \rho, g, h, l \)):
\[ l = 2h \]
\[ h = \frac{l}{2} = \frac{10}{2} = 5 \text{ cm} \]
Step 3: Final Answer:
The cylinder should be filled to a height of 5 cm.