Hydrostatic forces on a hinged gate

In summary, the conversation discusses finding the hydrostatic force on the hinge of a gate using an integral. The person asks for help with setting up the integral and another person suggests using a force balance on the fluid contained between the gate and two dashed lines. This method can quickly give the horizontal and vertical components of the force exerted by the gate on the fluid.
  • #1
member 428835
I'm trying to just find the hydrostatic force on the hinge of the gate in the image attached, and while not labeled, the thickness out of the page is ##w=5##. My thoughts are to set up this integral $$\vec{F}_p = \iint \rho g y \hat{n}\, dA = w \rho g \int_0^8 y \left( \frac{-dx\hat{i}+dy\hat{j}}{\sqrt{dx^2+dy^2}} \right)\, \sqrt{dx^2+dy^2}\\
= w \rho g \int_0^8 y \left(-dx\hat{i}+dy\hat{j} \right)\\
= w \rho g \int_0^8 y \left(y'(x)\hat{j}-1\hat{i} \right) \, dx\\
=5\cdot 64 \int_0^8 \left(9 + \frac{3 x}{4}\right)(\frac{3}{4}\hat{j}-\hat{i})\, dx\\
=-30720 \hat{i} + 23040 \hat{j}$$
where the magnitude of the last expression gives me the correct answer of the magnitude of ##\vec{F}_p=38,400## lbf. So when finding the force on the hinge I would say it is simply ##-30720 \hat{i} + 23040 \hat{j}## but the book says something else. Where am I going wrong?
 

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  • #2
joshmccraney said:
Where am I going wrong?

Some of the force on the gate is supported by the non hinged end (at A). Taking moments about point A is probably the quickest way to find the force on the hinge.
 
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  • #3
Thanks, your response makes perfect sense! Could you help me set up this integral if the water was instead below the gate?
 
  • #4
Problem in #1 can be solved in a few lines using basic trig and simple arithmetic ?
 
  • #5
Josh,

What Nidum is referring to is this: Draw a dashed horizontal line from point A to the left, and draw a dashed vertical line from point B vertically upward. Then consider the fluid contained in the region between the gate and the dashed lines. Do a force balance on this fluid in the horizontal and vertical directions. This will quickly give you the horizontal and vertical components of the force exerted by the gate on the fluid (and, by Newton's 3rd law, the fluid on the gate).
 

What is the definition of hydrostatic forces on a hinged gate?

Hydrostatic forces on a hinged gate refer to the pressure exerted by a fluid on a gate that is hinged at one end and free to move at the other end. This can occur in systems such as dams, locks, and flood barriers.

What factors affect the hydrostatic forces on a hinged gate?

The hydrostatic forces on a hinged gate are affected by the density of the fluid, the depth of the fluid, the shape and size of the gate, and the angle of the gate in relation to the fluid flow. These factors can impact the magnitude and direction of the forces acting on the gate.

How do hydrostatic forces on a hinged gate impact the stability of the gate?

Hydrostatic forces can cause a net torque on a hinged gate, which can affect its stability. If the forces are not balanced, the gate may open or close unintentionally, potentially causing damage to the gate or the surrounding structures.

How can engineers calculate the hydrostatic forces on a hinged gate?

Engineers can use the principles of fluid mechanics, specifically Pascal's law and Archimedes' principle, to calculate the hydrostatic forces on a hinged gate. They can also use computational fluid dynamics simulations to model the forces and their effects on the gate.

What are some real-world applications of hydrostatic forces on a hinged gate?

Hydrostatic forces on a hinged gate are crucial in the design and operation of various water management systems, such as flood barriers, dams, and canal locks. They are also important in the design of hydraulic equipment, such as gates in irrigation systems and water treatment plants.

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