Resultant Force Due to Hydrostatic Pressure

[MJay82]

I've got a question concerning the resultant force due to hydrostatic pressure. I understand how to calculate an object with uniform dimensions as depth (and pressure) increase. But I got thrown a serious curve ball on a Statics test.

The situation was such:

A downward facing isosceles triangular gate with base b and height a is hinged on the top at point O, which is a distance h below the surface of water. Calculate the force exerted on the back side of the gate to keep it closed.

I feel confident in finding the point that the resultant pressure force is applied at, but I got really confused because surface area of the gate is decreasing with depth while pressure is increasing with depth.

I see this equation here,
dp = [tex]\gamma[/tex]dh
Can I use this to find the magnitude of the resultant pressure force? I've got a final tomorrow and I feel that if only I could understand this one concept, I could work out the rest of my problems fairly easily. Thanks for any help.

 

[tiny-tim]

Hi MJay82!

Standard method: slice the gate into horizontal slices from depth h to h+dh, find the force on each slice, and integrate.

(and there's a hinge, so this is a rotational problem, so in this case you'd need the moment of the force on each slice, about the hinge)

 

[MJay82]

Ok - that's what I thought would be going on, thanks for the speedy reply!
I should have mentioned that I did understand the moment part. Using the scalar moment calculation (force x perpendicular distance), finding the force necessary to keep the gate closed is a cinch.

Let's see if I can maybe more clearly articulate what is baffling me, because although your answer let's me know that I'm at least thinking correctly, I can't conceptualize the changes.

You've got a pressure which is varying with depth - increasing as depth increases. And you've got an area which is also varying with depth, but is decreasing as depth increases.

I'm not familiar with double integration yet - is this a problem that would call for such, or since area and pressure are both varying with respect to depth, is it just a matter of relating both area and pressure in terms of one variable?

 

[tiny-tim]

I'm not familiar with double integration yet - is this a problem that would call for such, or since area and pressure are both varying with respect to depth, is it just a matter of relating both area and pressure in terms of one variable?

Forget about variables, just ask yourself how many d's are there?

In this case, there's only dh …

so you'll be summing over all the slices, which becomes a single integral ∫ … dh.

Just insert the moment of the force on the whole slice.

(If the pressure was also varying horizontally, then you'd need to slice-and-dice, and you'd have to sum over little squares of height dh and width dx … two d's, so a double integral. )

 

[alphy]

Thank you for your question. The situation you described is a classic example of a hydrostatic pressure problem in a statics context. In order to calculate the force exerted on the back side of the gate, we first need to understand the concept of resultant force due to hydrostatic pressure.

The resultant force due to hydrostatic pressure is the total force exerted on a submerged object by the surrounding fluid. This force is equal to the product of the pressure at a given depth and the surface area of the object at that depth. In other words, the force is a function of both the depth and the shape of the object.

In the case of your triangular gate, the surface area of the gate decreases with depth while the pressure increases. This means that the resultant force will also change as the gate is submerged deeper into the water.

To calculate the force exerted on the back side of the gate, you can use the equation you mentioned: dp = \gammadh. However, this equation only gives you the pressure at a given depth. To find the total force, you will need to integrate this equation over the surface area of the gate.

Alternatively, you can also use the concept of centroid to find the point at which the resultant force is applied. This point is known as the center of pressure and can be found by considering the distribution of pressure over the surface area of the gate. Once you have found the center of pressure, you can use it to calculate the magnitude and direction of the resultant force.

I hope this helps clarify the concept of resultant force due to hydrostatic pressure. Good luck on your final tomorrow!