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The problem is asking to determine the resultant hydrostatic force acting on the gate \(ABC\) due to water pressure. Here's a step-by-step breakdown to solve this:

### Given dataThe problem is asking to determine the resultant hydrostatic force acting on the gate \(ABC\) due to water pressure. Here's a step-by-step breakdown to solve this:

### Given 
- Gate width \( w = 1.0 \, \text{m} \)
- Water density \( \rho = 1.0 \, \text{Mg/m}^3 = 1000 \, \text{kg/m}^3 \)
- The depth of water at the top of the gate \(B\) is \(1.5 \, \text{m}\), and the depth at the bottom of the gate \(A\) is \(1.5 + 2 \times \sin 60^\circ\).
- Gate's angle \( \theta = 60^\circ \).

### Hydrostatic pressure equation:

The pressure at any depth \(h\) is given by:
\[
P = \rho g h
\]
where \(g = 9.81 \, \text{m/s}^2\) (acceleration due to gravity), and \(h\) is the depth below the water surface.

### Force due to hydrostatic pressure:

The total force \(F\) on a surface submerged in water can be calculated by integrating the pressure over the area:
\[
F = \int P \, dA = \rho g \int h \, dA
\]

However, for a plane surface (such as this gate), the force simplifies to:
\[
F = \rho g \bar{h} A
\]
where:
- \( \bar{h} \) is the depth to the centroid of the surface (average depth),
- \(A\) is the area of the gate submerged in water.

### Step 1: Calculate the depth to the centroid (\( \bar{h} \))

The gate is a rectangle inclined at \(60^\circ\) with a height of \(2 \, \text{m}\). The centroid of the gate (for pressure calculations) will be at half its height, so:

\[
\bar{h} = 1.5 + \frac{2}{2} \sin 60^\circ = 1.5 + 1 \times 0.866 = 2.366 \, \text{m}
\]

### Step 2: Calculate the area of the gate

The area \(A\) of the gate is:
\[
A = 2 \, \text{m} \times 1 \, \text{m} = 2 \, \text{m}^2
\]

### Step 3: Calculate the total hydrostatic force

Now we can calculate the resultant force using:
\[
F = \rho g \bar{h} A
\]
Substitute the values:
\[
F = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 2.366 \, \text{m} \times 2 \, \text{m}^2
\]
\[
F = 1000 \times 9.81 \times 2.366 \times 2 = 46393.32 \, \text{N}
\]

### Final Result:
The magnitude of the resultant hydrostatic force acting on the gate is approximately:
\[
F \approx 46.4 \, \text{kN}
\]

Would you like further details on specific steps or any related questions?

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Here are 5 related questions to explore this topic further:
1. How do we calculate the location of the resultant force (center of pressure) for an inclined plane?
2. What role does the angle of inclination play in calculating hydrostatic forces?
3. How would the hydrostatic force change if the gate were vertical instead of inclined?
4. How does the width of the gate influence the resultant force?
5. What is the difference between the force acting on a submerged horizontal surface versus a vertical surface?

**Tip**: Always ensure you correctly determine the centroid when dealing with hydrostatic forces, especially for inclined surfaces.

Determine the magnitude of the resultant force acting on the gate ABC due to the hydrostatic pressure. The gate has a width of 1.0 m. The density of water is 1.0 Mg/m^3.

Step-by-step solution to determine the hydrostatic force acting on an inclined gate due to water pressure. Concepts include calculating pressure, centroid depth, and total force. Suitable for advanced high school or undergraduate students.

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