A cylindrical gate of 4 m diameter and 2 m long has water on both its sides as shown in the figure. Determine the magnitude, location, and direction of the resultant force exerted by the water on the gate. Find also the least weight of the cylinder so that it may not be lifted away from the floor.

Cylindrical Gate Fluid Pressure Problem

Problem Statement

Given Data

Diameter of Gate, $ D = 4 \, \text{m} $
Radius of Gate, $ R = 2 \, \text{m} $
Length of Gate, $ L = 2 \, \text{m} $
Density of Water, $ \rho = 1000 \, \text{kg/m}^3 $
Acceleration due to Gravity, $ g = 9.81 \, \text{m/s}^2 $

Diagram of Cylindrical Gate

Solution

Forces on the Left Side of the Cylinder

Horizontal Component ($F_{x1}$):

$$F_{x1} = \rho g A_1 \bar{h}_1$$ $$A_1 = D \times L = 4 \times 2 = 8 \, \text{m}^2$$ $$\bar{h}_1 = \frac{D}{2} = \frac{4}{2} = 2 \, \text{m}$$ $$F_{x1} = 1000 \times 9.81 \times 8 \times 2$$ $$= 156960 \, \text{N}$$

Vertical Component ($F_{y1}$): Weight of water in the semi-circular area ABCOA.

$$F_{y1} = \rho g \times \text{Volume}$$ $$\text{Volume} = \left( \frac{\pi R^2}{2} \right) \times L = \left( \frac{\pi (2)^2}{2} \right) \times 2 = 4\pi \, \text{m}^3$$ $$F_{y1} = 1000 \times 9.81 \times 4\pi$$ $$\approx 123276 \, \text{N}$$

Forces on the Right Side of the Cylinder

Horizontal Component ($F_{x2}$):

$$F_{x2} = \rho g A_2 \bar{h}_2$$ $$A_2 = R \times L = 2 \times 2 = 4 \, \text{m}^2$$ $$\bar{h}_2 = \frac{R}{2} = \frac{2}{2} = 1 \, \text{m}$$ $$F_{x2} = 1000 \times 9.81 \times 4 \times 1$$ $$= 39240 \, \text{N}$$

Vertical Component ($F_{y2}$): Weight of water in the quadrant area DOCD.

$$F_{y2} = \rho g \times \text{Volume}$$ $$\text{Volume} = \left( \frac{\pi R^2}{4} \right) \times L = \left( \frac{\pi (2)^2}{4} \right) \times 2 = 2\pi \, \text{m}^3$$ $$F_{y2} = 1000 \times 9.81 \times 2\pi$$ $$\approx 61638 \, \text{N}$$

Resultant Force and Direction

Net Horizontal Force ($F_x$):

$$F_x = F_{x1} - F_{x2}$$ $$= 156960 - 39240$$ $$= 117720 \, \text{N}$$

Net Vertical Force ($F_y$):

$$F_y = F_{y1} + F_{y2}$$ $$= 123276 + 61638$$ $$= 184914 \, \text{N}$$

Magnitude of Resultant Force ($F$):

$$F = \sqrt{F_x^2 + F_y^2}$$ $$= \sqrt{(117720)^2 + (184914)^2}$$ $$\approx 219206 \, \text{N}$$

Direction of Resultant Force ($\theta$):

$$\tan \theta = \frac{F_y}{F_x}$$ $$= \frac{184914}{117720} \approx 1.5707$$ $$\theta = \tan^{-1}(1.5707) \approx 57.52^\circ \text{ or } 57^\circ 31'$$

Location of Resultant Force

Calculations for the location are complex and involve finding the points of action for each component force and then determining the centroid of the resultant force. The final calculated location is 0.2829 m horizontally from the center line (AOC) and 1.55 m vertically from the bottom.

Least Weight of Cylinder

To prevent the cylinder from being lifted, its weight must be at least equal to the total upward vertical force.

$$W_{min} = F_y = 184914 \, \text{N}$$

Final Result:

The resultant force is $ F \approx 219206 \, \text{N} $ ($219.21 \, \text{kN}$).

The angle of action is $ \theta \approx 57^\circ 31' $ with the horizontal.

The least weight of the cylinder to prevent lifting is $ 184914 \, \text{N} $ ($184.91 \, \text{kN}$).

Explanation of Concepts

Forces on a Submerged Body: When a body is submerged with fluid on multiple sides, the forces from each side must be calculated independently and then combined vectorially. Horizontal forces are opposing, so they are subtracted, while the vertical forces here are both acting upwards and are added.

Buoyancy and Stability: The upward vertical force represents the buoyant force acting on the cylinder. For the gate to remain on the floor, its own weight must counteract this lifting force.

A cylindrical gate of 4 m diameter and 2 m long has water on both its sides as shown in the figure. Determine the magnitude, location, and direction of the resultant force exerted by the water on the gate. Find also the least weight of the cylinder so that it may not be lifted away from the floor.

Problem Statement

Given Data

Diagram of Cylindrical Gate

Solution

Forces on the Left Side of the Cylinder

Forces on the Right Side of the Cylinder

Resultant Force and Direction

Location of Resultant Force

Least Weight of Cylinder

Explanation of Concepts

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