## Understanding Hydrostatic Force on Plane Surface

The hydrostatic force on a plane surface refers to the force exerted by a fluid at rest. This force is resultant from the fluid pressure multiplied by the area over which the pressure is distributed. Before delving into the concept, it becomes incumbent upon us to have a robust understanding of a few basic terms.

Pressure: It is the force applied perpendicular to the surface of an object per unit area where the force is spread.

Fluid: A fluid is any substance that can flow, for instance, liquids and gases.

### Concept of Hydrostatic Force on Plane Surface

The pressure in a liquid increases proportionally with its depth. Hence, pressure exerted by a liquid column of uniform cross-sectional area is given by \( P = \rho · g · h \), where \( P \) is the pressure, \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid column above the point in question.

Therefore, the hydrostatic force acting on a plane surface, submerged in a static liquid can be calculated by integrating the pressure over the entire submerged area.

For instance, if a rectangular dam with a width \( b \) and height \( h \) is submerged to a depth \( d \) beneath the surface of a lake, the hydrostatic force \( F \) on the dam can be calculated using the formula: \( F = \frac{1}{2} · \rho · g · b · h^2 · (d + \frac{h}{3}) \).

It's noteworthy to remember that for plane surfaces perpendicular to the liquid surface, the hydrostatic force is a function of only the depth of the centroid of the area under the liquid surface, the area of the surface, and the fluid's density.

### Influencing Factors on Hydrostatic Force on Plane Surface

Several factors influence the magnitude and distribution of hydrostatic force on a plane surface. They are principally,

- The depth of the centroid of the submerged area beneath the liquid surface, which directly influences the hydrostatic pressure acting on the surface.
- The area and shape of the submerged surface, which determine the total distribution of pressure. Note, however, the shape of the surface doesn’t affect the resultant hydrostatic force.
- The density of the liquid, as a higher density fluid exerts more pressure.

Note that gravity influences the hydrostatic force indirectly by affecting the pressure in terms of \( P = \rho · g · h \).

For example, the hydrostatic force on a circular gate of a dam (of radius \( r \) and density \( \rho \)) at a depth \( d \) beneath the lake surface can be calculated by \( F = \pi · \rho · g · r^2 · (d + \frac{2r}{3}) \)

## Hydrostatic Force on Different Plane Surfaces

Different types of surfaces submerged in a static fluid encounter varying distributions and magnitudes of hydrostatic force. These variations are primarily due to their geometry, orientation, and the depth of the liquid above the centroid of the submerged area. To gain a thorough understanding of the hydrostatic force exerted on plane surfaces, let's now discuss these surfaces in three different scenarios: vertical, inclined, and submerged plane surfaces.

### Hydrostatic Force on Vertical Plane Surface

When a plane surface is vertical, the variations in pressure across the surface lead to a non-uniform distribution of hydrostatic force. The pressure at any point on the surface is proportional to the depth of that point below the liquid surface, a principle articulated as \( P = \rho · g · h \) where \( P \) is pressure, \( \rho \) is fluid density, \( g \) is gravity, and \( h \) is depth below the surface.

The total hydrostatic force \( F \) acting on the surface can be found by integrating the pressure over the area of the surface. For a rectangular surface submerged vertically in a liquid with height \( h \) and width \( b \), the total hydrostatic force \( F \) is given by \( F = \frac{1}{2}\rho · g · b · h^2 \).

For a rectangular surface of height 5m and width 3m, submerged vertically in water with a density of 1000 kg/m³, the hydrostatic force is given by \( F = \frac{1}{2} · 1000 · 9.81 · 3 · (5)^2 = 367.5 kN \).

### Hydrostatic Force on Submerged Plane Surfaces

A submerged plane surface refers to any surface that is completely immersed in a fluid and is not in contact with the free surface of the liquid. Just like vertical surfaces, there will be non-uniform distribution of pressure which manifests from variations in depth.

Considering a surface submerged in a liquid with area \( A \) at a depth \( h \), the center of pressure on an immersed surface is the point where the total sum of hydrostatic force is assumed to act. The depth \( y_c \) of the centre of pressure from the surface of the fluid is given by \( y_c = \frac{I_G}{A·h} + h \) where \( I_G \) is the second moment of area.

The total hydrostatic force on a submerged plane surface, be it horizontal, vertical, or inclined, is the same and can be calculated as \( F = P_{cp} · A \), where \( P_{cp} \) is pressure at the centroid."

For an equilateral triangular plate of side 3m, submerged in water with vertex at the surface and the base parallel to and 3m below the water surface, the hydrostatic force can be calculated by first finding the pressure at the centroid (\( P_{cp} = \rho · g · h \)) and then multiplying it with the area of the triangle.

### Hydrostatic Force on Inclined Plane Surface

An inclined plane surface is one that is tilted at an angle other than 90 degrees with respect to the surface of the static liquid. Calculating the hydrostatic force on an inclined surface involves accounting for both the area of the surface and the depth of the centroid below the surface.

For a rectangular surface of width \( b \) and height \( h \), inclined at an angle \( \theta \), the total hydrostatic force \( F \) is given by \( F= \frac{1}{2}\rho · g · b · h^2 · \cos{\theta} \).

A rectangular gate 1m wide and 2m high, installed at a 45 degree angle in a tank filled with oil of relative density 0.9 would experience a hydrostatic force given by \( F = \frac{1}{2} · 0.9 · 1000 · 9.81 · 1 · (2)^2 · \cos{(45)} = 13.85 kN \)

## Mathematical Approach towards Hydrostatic Force on Plane Surface

To precisely determine the hydrostatic force exerted on a plane surface submerged in a liquid, one must delve into the mathematical realm. The mathematical approach provides essential tools to calculate both the magnitude and the line of action of the resultant hydrostatic forces. The approach predominantly hinges upon the basic concepts of calculus and mechanics as the extent of hydrostatic force needs to be calculated as an integral over the submerged area.

### Hydrostatic Force on Plane Surface Formula

The mathematical formulation for calculating the hydrostatic force on a plane surface relies on the principles of fluid mechanics. The formula conjuncts some fundamental parameters that include the fluid's density, the acceleration due to gravity, and the depth and area of the submerged surface.

The pressure \( P \) at any point in a fluid at rest is given by \( P = \rho · g · h \), where \( \rho \) is the density of the fluid, \( g \) is the acceleration due to gravity and \( h \) is the height of the fluid column above the point.

The hydrostatic force \( F \) on a plane surface is the integral of pressure over the area \( A \) of the surface, typically computed as \( F = \int_A P \,dA \). However, this integration can be simplified as \( F = P_{cp} · A \), where \( P_{cp} \) is the pressure at the centroid of the plane surface.

The formula for calculating pressure at the centroid is \( P_{cp} = \rho · g · h_c \), where \( h_c \) is the vertical depth of the centroid of the submerged surface from the free surface of the liquid.

The total hydrostatic force \( F \) on the plane surface then simplifies to \( F = \rho · g · h_c · A \), putting \( P_{cp} \) in place.

### Hydrostatic Force on Plane Surface Examples

Examples always serve to illuminate theoretical principles. Let's walk through a few to get a grasp of how to apply the above-discussed formulas.

**Example 1:** Consider a rectangular gate 6m wide and 4m high, fixed along the top edge with its plane vertical. The gate is holding back water on one side. Figure out the resultant hydrostatic force on the gate and its line of action.

Here, the total pressure on the centroid \( P_{cp} \) is \( P_{cp} = \rho · g · h_c = 1000 · 9.81 · 2 = 19620 Pa \).

The area of the gate \( A \) is \( A = b · h = 6 · 4 = 24 m^2 \).

The resultant hydrostatic force \( F \) is \( F = P_{cp} · A = 19620 · 24 = 470880 N = 471 kN \).

**Example 2:** Compute the hydrostatic force on a circular gate of diameter 3m, submerged in oil with relative density 0.8, with its centre 2.5m beneath the oil surface.

The radius of the gate (\( r \)) is half the diameter i.e. \( r = 1.5m \).

As the gate is circular, the area \( A \) can be calculated as \( A = \pi · (r)^2 = \pi · (1.5)^2 = 7.07 m^2 \).

The pressure at the centroid \( P_{cp} \) is \( P_{cp} = \rho · g · h_c = 800 · 9.81 · 2.5 = 19620 Pa \).

Thus, the resultant hydrostatic force \( F \) is \( F = P_{cp} · A = 19620 · 7.07 = 138674 N = 138.7 kN \).

Remember, you need to understand the principles and formulae thoroughly to adeptly calculate the hydrostatic force in diverse circumstances.

## Hydrostatic Force on Plane Surface - Key takeaways

- Hydrostatic force on a plane surface refers to the force exerted by a fluid at rest, resultant from the multiplication of fluid pressure and the area over which the pressure is distributed.
- Factors influencing hydrostatic force include the depth of the centroid of the submerged area, the area and shape of the submerged surface, and the fluid's density.
- Hydrostatic force on different plane surfaces such as vertical, submerged, and inclined surfaces differs due to their geometry, orientation, and the depth of the liquid above the centroid of the submerged area.
- The formula for calculating hydrostatic force on a plane surface is \( F = P_{cp} · A \), where \( P_{cp} \) is the pressure at the centroid of the plane surface and \( A \) is the area of the surface. The pressure \( P_{cp} \) at the centroid is calculated by \( P_{cp} = \rho · g · h_c \), where \( \rho \) is the fluid's density, \( g \) is acceleration due to gravity and \( h_c \) is the vertical depth of the centroid of the submerged surface from the surface of the fluid.
- Different examples explained how the formulae are applied to calculate hydrostatic force on different plane surfaces in diverse circumstances.

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