To solve this problem, we need to understand the forces acting on the gate due to the water pressure and how these forces relate to the force $F$ applied to keep the gate closed.

Step-by-step approach:

1. Hydrostatic Force on the Gate: The force exerted by the water on the gate is due to the hydrostatic pressure, which increases with depth. The hydrostatic pressure at a depth $h$ below the water surface is given by:

   $P(h) = \rho g h$

   Where:

   • $\rho$ is the density of water ($1000 \, \text{kg/m}^3$).
   • $g$ is the acceleration due to gravity ($9.81 \, \text{m/s}^2$).
   • $h$ is the depth below the water surface.

2. Total Force on the Gate: The hydrostatic force $F_{w}$ on a vertical surface of height $H$ submerged in water can be found by integrating the pressure across the height of the gate. The total force on the gate is:

   $F_{w} = \rho g \cdot A \cdot h_{\text{center of pressure}}$

   Where $A$ is the area of the gate (height $H \times$ width $1 \, \text{m}$), and $h_{\text{center of pressure}}$ is the depth to the center of pressure (which for a vertical gate is at a depth of $\frac{h + H}{2}$).

3. Moment Calculation: The force $F_w$ acts at the center of pressure. To keep the gate in a closed position, the applied force $F$ must create an equal and opposite moment to counter the moment due to the water force about the hinge.

   The moment due to the water force is:

   $M_{w} = F_{w} \cdot \text{distance from hinge}$

   The applied force $F$ is applied at a certain distance from the hinge. If we know the location where $F$ is applied, we can write the moment balance equation:

   $F \cdot \text{distance from hinge} = M_{w}$

4. Solving for the Applied Force $F$: Rearranging the moment equation, we can solve for $F$:

   $F = \frac{M_{w}}{\text{distance from hinge}}$

Could you provide a sketch or describe more details about where the force $F$ is applied and the exact hinge location? This information is necessary to calculate the applied force $F$.

Would you like me to clarify any particular step, or provide further details? Here are some related questions that might help:

1. How does the depth of water affect the total force acting on the gate?
2. What is the significance of the center of pressure in calculating hydrostatic forces?
3. How do you calculate moments when multiple forces are acting on an object?
4. Why does hydrostatic pressure increase linearly with depth?
5. How does the width of the gate affect the total force?

Tip: Always remember that the center of pressure is typically lower than the centroid for submerged surfaces because pressure increases with depth.