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

Concept: Hydrostatic force: \[ F = P \cdot A = \rho g h \cdot A \] For vertical surfaces, average pressure is taken.

Step 1: Force on bottom.
Base area: \[ A = 10 \times 10 = 100 \, \text{cm}^2 \] \[ F_{bottom} = \rho g h \cdot 100 \]

Step 2: Force on sides.
Total side area: \[ A_{side} = 4 \times (10 \times h) = 40h \] Average pressure on side: \[ P_{avg} = \frac{\rho g h}{2} \] \[ F_{side} = P_{avg} \cdot A_{side} = \frac{\rho g h}{2} \cdot 40h = 20 \rho g h^2 \]

Step 3: Equate forces.
\[ \rho g h \cdot 100 = 20 \rho g h^2 \] \[ 100h = 20h^2 \Rightarrow h = 5 \, \text{cm} \]