Hydrostatic Force on a Plane Surface

In summary, the derivation for Hydrostatic force on a plane surface involves integrating the differential force over the entire area of the surface. This can be represented by a symbol A at the base of the integral sign, indicating integration over the entire area of the plate, which could be a double integral. The dA in the integral could also be represented as wdy, where w is the width of the plate in the z direction. This integration allows for the summation of individual dA's on the surface, even if they are not on the same line, leading to the calculation of the total surface area.
  • #1
Arc_Unbated
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Homework Statement



I am confused with the derivation of Hydrostatic force on a plane surface. What is confusing me is, how can the integral of the differential force get the resultant force for the entire area of the surface? The differential force is as follows:
dF = ρgh dA

and the magnitude of the resultant force can be obtained by integrating the differential force over the whole area:
dF = ρg∫h dA

Whats bothering me is, should this not be a double integral, to integrate in the x direction, and then the y direction for the entire surface? I don't understand how integrating with respect to dA will add up all the dA's to give the total surface area.

The full derivation is on this page:
https://ecourses.ou.edu/cgi-bin/eBook.cgi?doc=&topic=fl&chap_sec=02.3&page=theory

Basically, why does the integral of ρgh dA give the total area? I can understand that it would sum up the individual dA's on a single line on the surface, but I can't understand how it adds up all the dA's that are not on the same line, but on another position on the plane.

Thanks for your help
 
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  • #2
In the derivation there is a symbol A at the base of the integral sign. That denotes that the integration is over the entire area of the plate which might well be a double integral. It's merely a shorthand notation. dA could also have been written as wdy where w is the width of the plate (z direction). In that case it would be a single integral.
 
  • #3
LawrenceC said:
In the derivation there is a symbol A at the base of the integral sign. That denotes that the integration is over the entire area of the plate which might well be a double integral. It's merely a shorthand notation. dA could also have been written as wdy where w is the width of the plate (z direction). In that case it would be a single integral.

thank you very much that answeres my question. Really appreciate that, i was getting frustrated trying to follow the logic bit by bit through the derivation in my notes, and then I got stuck there and thought i would just have to accept it. Now i understand, thanks again
 

Related to Hydrostatic Force on a Plane Surface

1. What is hydrostatic force on a plane surface?

Hydrostatic force on a plane surface is the force exerted by a fluid at rest on a flat surface, perpendicular to the surface. It is caused by the weight of the fluid above the surface and the pressure of the fluid acting on the surface.

2. How is hydrostatic force calculated on a plane surface?

The hydrostatic force on a plane surface can be calculated by multiplying the pressure at the centroid of the surface by the area of the surface. The pressure can be found by using the equation P = ρgh, where ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth of the fluid above the surface.

3. What factors affect the hydrostatic force on a plane surface?

The hydrostatic force on a plane surface is affected by the density of the fluid, the depth of the fluid above the surface, and the shape and orientation of the surface. The force also increases with increasing surface area.

4. How is the direction of hydrostatic force determined on a plane surface?

The direction of hydrostatic force on a plane surface is always perpendicular to the surface, pointing away from the surface and towards the direction of the fluid's weight. This direction is known as the normal direction.

5. What are some real-life applications of hydrostatic force on a plane surface?

Hydrostatic force on a plane surface is used in many practical applications, such as in dams, retaining walls, and ship design. It is also important in understanding and predicting the behavior of fluids in pipes, tanks, and other containers.

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