# Fluid Mechanics: Hydrostatic Pressure on a gate with hinges

• Feodalherren
In summary, the water exerts a 270Nm torque on a horizontal strip of the gate if the strip extends from y to y+dy.
Feodalherren

Fluid Mechanics

## The Attempt at a Solution

The pressure of the air is constant so the pressure will act as if it's a concentrated force on the center of gravity of the gate.

F = PA = (10,000Pa)(.3)(.6)=1800N

The moment about B = 1800N (.15m) = 270Nm.

Now we know that the water has to exert a 270Nm torque to make the gate move.

This is where I'm stuck. I'm not sure how to find the torque that the water exerts on the gate. Depending on the ratio of h to the height of the door, .3, it seems to me that the concentrated force of the water might land above the door. So it seems to me like I can't use the same method. I feel like I need to integrate and add up all the small moments across the whole door.

So the integral would be something like

integrating from B to .3

M = ∫ γ(volume of water)dy

but I don't know anything else except the height for the water...

Feodalherren said:
I feel like I need to integrate and add up all the small moments across the whole door.

So the integral would be something like

integrating from B to .3

M = ∫ γ(volume of water)dy

but I don't know anything else except the height for the water...

OK. You have the right idea. If you let y denote depth below point B, how would you express the force from the water on a horizontal strip of the gate if the strip extends from y to y+dy?

Feodalherren
Hmm I'm not sure. I guess that's kind of where I'm stuck.

The pressure would depend on h and I'm not sure how to relate h to the length of the gate.

From the picture below, what is the depth, H, of the strip below the surface of the water in terms of h, y, and .30 m?

#### Attachments

• hydrogate.png
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Feodalherren
H= h + y - .30

?

Feodalherren said:
H= h + y - .30

?
Yes. So, you can express the force on the strip in terms of H, and therefore in terms of y and h.

Feodalherren
Ok so dA= .6 dy

The force exerted by the water on a differential of the area is

Fw = P(.6)dy

The pressure of the water:

ϒ = 9790 N / m^3

P= ϒ(h+y-.3)

then the force becomes (ignoring direction since we know it's opposite to the direction of the force of the air).

Fw(total) = ∫ ϒ(h+y-.3)(.6)dy

integrate from 0 to .3

is that correct?

Feodalherren said:
The pressure of the water:

ϒ = 9790 N / m^3

P= ϒ(h+y-.3)

OK, I guess you're using 9.79 m/s2 for g.

then the force becomes (ignoring direction since we know it's opposite to the direction of the force of the air).

Fw(total) = ∫ ϒ(h+y-.3)(.6)dy

integrate from 0 to .3

is that correct?

This would yield the force, but you want the torque.

Feodalherren
The number came out of my book, Fluid Mechanics by Frank White 5th ed.

Yeah I just wanted to check my logic. Okay so if that's the total force then the total torque must just be the total force multiplied by the distance

T = F y

where y is the distance from B to the point of application of the force.

so

T = ∫ ϒ(h+y-.3)(.6)y dy

correct?

Looks good.

Feodalherren
Thank you so much! I got the correct answer!

Great! Good work.

## What is fluid mechanics?

Fluid mechanics is the study of how fluids (liquids and gases) behave and interact with their surroundings. It involves understanding their physical properties, such as density and viscosity, and how they flow and exert forces.

## What is hydrostatic pressure?

Hydrostatic pressure is the pressure exerted by a fluid at rest due to the weight of the fluid above it. It is a result of the force of gravity acting on the fluid and can be calculated using the equation P = ρgh, where P is the pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the height of the fluid column.

## How does hydrostatic pressure affect a gate with hinges?

In a gate with hinges, hydrostatic pressure will exert a force on the gate, causing it to push against the hinges. This force is directly proportional to the density of the fluid and the height of the fluid column above the gate. As the height of the fluid increases, so does the force on the gate, which can potentially cause damage if the gate is not properly designed to withstand it.

## What factors can affect hydrostatic pressure on a gate with hinges?

The two main factors that can affect hydrostatic pressure on a gate with hinges are the density of the fluid and the height of the fluid column above the gate. Other factors that may also play a role include the shape and size of the gate, the strength and positioning of the hinges, and the overall design of the gate and its surroundings.

## How can hydrostatic pressure on a gate with hinges be managed?

To manage hydrostatic pressure on a gate with hinges, it is important to carefully consider the design and materials used. The gate should be able to withstand the expected pressure, and the hinges should be strong enough to support the weight of the gate and the pressure exerted by the fluid. Proper maintenance and regular inspections can also help identify any potential issues and prevent damage from occurring.

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