Question

A square gate of size \(1\,\text{m} \times 1\,\text{m}\) is hinged at its mid-point. A fluid of density \(\rho\) fills the space to the left of the gate. The force \(F\) required to hold the gate stationary is: 

• A. \(\dfrac{\rho g}{3}\)
• B. \(\dfrac{\rho g}{2}\)
• C. \(6\rho g\)
• D. \(\dfrac{\rho g}{8}\)

Hint:

For fluid pressure problems: \[ p=\rho g h \] Always take moments about the hinge point.

Answer 1

The Correct Option is A

Step 1: Pressure at depth \(y\): \[ p=\rho g y \]
Step 2: Force on an elemental strip: \[ dF = \rho g y \cdot dy \]
Step 3: Total torque about hinge: \[ \tau = \int_0^1 \rho g y \cdot y\,dy = \rho g \int_0^1 y^2 dy = \frac{\rho g}{3} \]
Step 4: Balancing torque gives required force: \[ F=\frac{\rho g}{3} \]