Why must we use integral to find the resultant force?

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Homework Help Overview

The discussion revolves around understanding the application of integrals in calculating hydrostatic forces on plane surfaces, specifically in the context of a gate compared to a slanted plane. Participants are exploring the differences in approach and reasoning behind the use of integrals versus direct formulas.

Discussion Character

  • Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants question why a direct formula cannot be used for the problem and express confusion regarding the transition from a double integral to a single integral. There is also a discussion about the implications of viewing the gate from different perspectives and how that affects the calculations.

Discussion Status

The conversation is ongoing, with participants providing insights into the relationship between the gate and slanted planes. Some guidance has been offered regarding the constants involved in the equations, but there is still exploration of the underlying concepts without a definitive consensus.

Contextual Notes

Participants are navigating the complexities of hydrostatic pressure and the geometry of the surfaces involved, including the significance of angles and constants in their calculations. There is an emphasis on understanding the differences in setup between vertical gates and slanted planes.

EastWindBreaks
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Homework Statement


Hydrostatic force on a plane surface ex:
upload_2017-11-13_3-15-19.png


Hydrostatic force on a gate:
upload_2017-11-13_3-12-35.png


Homework Equations



The Attempt at a Solution


why can't we just use the formula in the red box above for problem 3.57?
instead I have to use integral
upload_2017-11-13_3-38-14.png

upload_2017-11-13_3-17-44.png

upload_2017-11-13_3-17-14.png

I am confused, how does this gate different from a slanted plane. if I look at the gate from front view, i feel like they are just the same thing. How does
upload_2017-11-13_3-38-24.png
a double integral with respect to area (dhdb) became a single integral with respect to dh?
 

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oh! I can use it but because its the top view, we have to use 5 meter for h_c, which is the depth that affects the pressure.
 
ok, I have one question left, how does
upload_2017-11-13_3-38-24-png.png
a double integral with respect to area (dhdb) became a single integral with respect to dh?
 

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EastWindBreaks said:
double integral with respect to area (dhdb) became a single integral with respect to dh
Because the width (the b variable in dbdh) is constant.
EastWindBreaks said:
how does this gate different from a slanted plane.
It doesn't really. Just bear in mind that the gates are vertical, so in the equation in the red box you would put θ=π/2. On the other hand, the area is not WD. To express the answer in terms of W and D you need to make reference to the 15° angle.
With those adjustments, I think you will find the two expressions are the same.
 
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haruspex said:
Because the width (the b variable in dbdh) is constant.

It doesn't really. Just bear in mind that the gates are vertical, so in the equation in the red box you would put θ=π/2. On the other hand, the area is not WD. To express the answer in terms of W and D you need to make reference to the 15° angle.
With those adjustments, I think you will find the two expressions are the same.
oh yeah, I got it now, thank you!
 

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