Angle

45°

Water Height

0.95

m

Gate Weight

10.0

kN

Units:

Show distances

Directions

Details

About

A gate, which is hinged at the bottom, is partially submerged under water, and a cable holds the gate closed. Use the sliders to set the angle of the gate, the weight of the gate, and the water height. Use the buttons to change the units from klb_{F} and ft (US customary units) to kN and m (SI units). Check the "show distances" box to display distances. The simulation displays the cable tension needed to support the gate.

This simulation determines the cable tension necessary to support a gate partially submerged under water. The gate is \( a_{2} \) meters long, and the distance from the hinge to top of the water along the gate is \( a_{1} \) meters. The vertical distance down from the top of the liquid is \( h \), and \( y \) is the distance along the gate, starting from the water's surface, such that \( h = h_{1} \) at the bottom of the container, and \( y = a_{1} \) at the hinge:
The magnitude of the resultant force due to the water is found by summing the differential forces \( dF = \gamma \, h \, dA \) over the entire surface:
$$
[1] \quad F_{R} = \int_{A} \gamma \, h \, dA = \int_{A} \, \gamma \, y \, \mathrm{sin ( \theta )} \, dA
$$
where \( F_{R} \) is the resultant force (N), \( \gamma \) is the specific weight of water (9.807 kN/m^{3}), \( h = y \, \mathrm{ sin ( \theta ) } \) is the vertical distance from the top of the water to any point in the water (m), \( \theta \) is the angle of the gate (degrees), \( dA = b \, dy \) is differential area of the gate (m^{2}), and \( b \) is the width of the gate (m). Note that the specific weight \( \gamma \) is the specific gravity \( \rho \) times the acceleration of gravity \( g \). The total area of the gate that is in contact with water is \( A_{gate} = b \, a_{1} \). This integral is from \( y = 0 \) at the top of the water level to \( y = a_{1} \) at the hinge. Since \( \gamma \) is constant, and for a fixed value of \( \theta \), the resultant force becomes:
$$
[2] \quad F_{R} = \gamma \, \mathrm{sin(\theta)} \int_{A} y \, dA
$$
The integral \( \int_{A} y \, dA \) is:
this is then equal to:
$$
[4] \quad \int_{A} y \, dA = \frac{b}{2} a^{2}_{1} = A_{gate} \frac{a_{1}}{2}
$$
the resultant force is then:
$$
[5] \quad F_{R} = \gamma \, \mathrm{sin( \theta )} A_{gate} \frac{a_{1}}{2}
$$
The sum of the moments around the hinge is equal to the moment of the resultant force at the y coordinate \( y_{R} \). Note that moment is proportional to the distance from the hinge to location of the force:
$$
[6] \quad F_{R} ( a_{1} - y_{R} ) = \int_{A} \gamma \, y \, (a_{1} - y) \, \mathrm{sin(\theta)} \, dA = \gamma \, \mathrm{sin(\theta)} \int_{0}^{a_{1}} y \, (a_{1} - y) \, b \, dy
$$
therefore,
$$
[8] \quad F_{R} ( a_{1} - y_{R} ) = \gamma \, \mathrm{sin(\theta)} \frac{a_{1}^{2}}{6} A_{gate}
$$
and substituting equation [5] into equation [8] and simplifying:
$$
[9] \quad y_{R} = \frac{ 2 \, a_{1} }{ 3 }
$$
That is, the resultant force is located 1/3 of the distance from the hinge to the water level along the gate (because \( y_{R} \) is the distance of the moment along the gate starting from the water's surface). A moment balance determines the tension \( T \) (kN) of the cable that is holding up the gate:
$$
[10] \quad \sum M_{i} = 0 = F_{R} \frac{a_{1}}{3} + W_{gate} \frac{a_{2}}{2} \mathrm{cos(\theta)} - T \, a_{2} \, \mathrm{sin(\theta)}
$$
where \( M_{i} \) is moment \( i \) and \( W_{gate} \) is the weight of the gate, which has its moment located at the center of the gate. The tension is then:
$$
[11] \quad T = \frac{ F_{R} \frac{ a_{1} }{ 3 } + W_{gate} \frac{ a_{2} }{ 2 } \mathrm{cos(\theta)} }{ a_{2} \, \mathrm{sin(\theta)} }
$$

This simulation was created in the Department of Chemical and Biological Engineering, at University of Colorado Boulder for LearnChemE.com by Rachael Baumann, Alex Jimenez, and Neil Hendren under the direction of Professor John L. Falconer. Address any questions or comments to learncheme@gmail.com. All of our simulations are open source, and are available on our LearnChemE Github repository.

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