A gate having a quadrant shape of radius of 3 m is positioned as shown in the figure below. Find the resultant force due to water per metre length of the gate. Find also the angle at which the total force will act.

Diagram of the Gate

Solution

The resultant force is found by calculating the horizontal and vertical components of the force separately.

1. Horizontal Force (\(F_x\))

The horizontal force is the total pressure on the vertical projection of the curved surface. This projection is a rectangle of height R = 3 m.

$$ A_x = R \times b $$ $$ A_x = 3 \times 1 = 3 \, \text{m}^2 $$

The depth of the centroid of this projected area is:

$$ \bar{h}_x = \frac{R}{2} $$ $$ \bar{h}_x = \frac{3}{2} = 1.5 \, \text{m} $$

Now, calculate the horizontal force component.

$$ F_x = \rho g A_x \bar{h}_x $$ $$ F_x = 1000 \times 9.81 \times 3 \times 1.5 $$ $$ F_x = 44145 \, \text{N} $$

2. Vertical Force (\(F_y\))

The vertical force is the weight of the water in the volume directly above the curved surface.

$$ V = \left( \frac{\pi R^2}{4} \right) \times b $$ $$ V = \left( \frac{\pi \times 3^2}{4} \right) \times 1 $$ $$ V = \frac{9\pi}{4} \approx 7.0686 \, \text{m}^3 $$

Now, calculate the vertical force component.

$$ F_y = \rho g V $$ $$ F_y = 1000 \times 9.81 \times 7.0686 $$ $$ F_y \approx 69343 \, \text{N} $$

3. Resultant Force (\(F\))

The resultant force is the vector sum of the components.

$$ F = \sqrt{F_x^2 + F_y^2} $$ $$ F = \sqrt{(44145)^2 + (69343)^2} $$ $$ F = \sqrt{1948780025 + 4808445049} $$ $$ F = \sqrt{6757225074} $$ $$ F \approx 82202 \, \text{N} $$

4. Angle of Action (\(\theta\))

The angle the resultant force makes with the horizontal is found using trigonometry.

$$ \tan \theta = \frac{F_y}{F_x} $$ $$ \tan \theta = \frac{69343}{44145} $$ $$ \tan \theta \approx 1.5708 $$ $$ \theta = \tan^{-1}(1.5708) $$ $$ \theta \approx 57.52^\circ \text{ or } 57^\circ 31' $$

Explanation of Concepts

Horizontal Component (\(F_x\)): The horizontal force on any curved surface is equal to the force that would be exerted on its vertical projection. For a quadrant gate like this, the projection is a rectangle with a height equal to the radius.

Vertical Component (\(F_y\)): The vertical force is equal to the weight of the fluid volume directly above the curved surface. For this gate, that volume is a quarter-cylinder with a length equal to the gate's width.

Resultant Force: Since the horizontal and vertical forces are perpendicular, the total resultant force can be found using the Pythagorean theorem, and its direction can be determined with basic trigonometry.