Name: hydrostatic force on dam walls with integration
Uploaded: 2026-01-12T12:24:31.000Z
Duration: 1 min 30 s
Channel: JoVE

Ciśnienie hydrostatyczne powstaje z masy płynu w spoczynku i rośnie proporcjonalnie wraz z głębokością.Dla powierzchni poziomej zanurzonej na stałej głębokości ciśnienie jest stałe i równe iloczynowi gęstości płynu, przyspieszenia grawitacyjnego oraz głębokości.Jednak dla powierzchni pionowej, takiej jak ściana tamy, każdy punkt leży na innej głębokości, co powoduje wzrost ciśnienia wraz z głębokością i niejednorodny rozkład sił.Aby obliczyć całkowitą siłę hydrostatyczną z tego zmiennego ciśnienia, zanurzona część dzieli się na cienkie poziome pasy. Każdy pas leży na określonej głębokości i ma powierzchnię równą szerokości pomnożonej przez niewielką grubość pionową.Ciśnienie na każdym pasku równa się gęstości masy płynu pomnożonej przez głębokość pod powierzchnią, a siła działająca na ten pasek równa się ciśnieniu pomnożonemu przez powierzchnię.Sumowanie tych sił na wszystkich pasach przybliża całkowitą siłę za pomocą sumy Riemanna. Gdy grubość paska zbliża się do zera, suma staje się całką określoną.Ta całka wyraża całkowitą siłę hydrostatyczną jako ciśnienie pomnożoną przez funkcję szerokości, całkowitą na głębokości zanurzonej.Ta metoda jest niezbędna w inżynierii lądowej do projektowania wydajnych konstrukcji oporowych.

zastosowania całkowania do wyznaczania ciśnienia hydrostatycznego

Applications of Integration to Find Hydrostatic Pressure

Si&#x142;a hydrostatyczna to ca&#x142;kowita si&#x142;a, z jak&#x105; ciecz w spoczynku dzia&#x142;a na powierzchni&#x119;. Dla poziomej powierzchni zanurzonej na sta&#x142;ej g&#x142;&#x119;boko&#x15B;ci ci&#x15B;nienie jest sta&#x142;e i obliczane jako iloczyn g&#x119;sto&#x15B;ci cieczy, przyspieszenia ziemskiego i g&#x142;&#x119;boko&#x15B;ci. W przypadku pionowej &#x15B;ciany zapory zanurzonej w wodzie si&#x142;a ta nie rozk&#x142;ada si&#x119; r&#xF3;wnomiernie ze wzgl&#x119;du na wzrost ci&#x15B;nienia wraz z g&#x142;&#x119;boko&#x15B;ci&#x105;. Ta zmienno&#x15B;&#x107; wynika z sumarycznego ci&#x119;&#x17C;aru wody nad ka&#x17C;dym punktem. Ca&#x142;kowanie s&#x142;u&#x17C;y do uwzgl&#x119;dnienia ci&#x105;g&#x142;ej zmiany ci&#x15B;nienia wzd&#x142;u&#x17C; zanurzonej powierzchni, co pozwala na dok&#x142;adne obliczenie ca&#x142;kowitej si&#x142;y hydrostatycznej.Ci&#x15B;nienie w cieczy w spoczynku ro&#x15B;nie liniowo wraz z g&#x142;&#x119;boko&#x15B;ci&#x105;. Na g&#x142;&#x119;boko&#x15B;ci h ci&#x15B;nienie P jest okre&#x15B;lone wzorem:\begin{equation*}P=\rho gh\end{equation*}&#xA0;gdzie &#x3C1; to g&#x119;sto&#x15B;&#x107; cieczy, a g to przyspieszenie grawitacyjne.Aby obliczy&#x107; si&#x142;&#x119; ca&#x142;kowit&#x105;, zanurzon&#x105; powierzchni&#x119; dzieli si&#x119; na niesko&#x144;czenie cienkie poziome paski. Ka&#x17C;dy pas na g&#x142;&#x119;boko&#x15B;ci h ma szeroko&#x15B;&#x107; w(h) i wysoko&#x15B;&#x107; r&#xF3;&#x17C;niczkow&#x105; dh, wi&#x119;c jego pole wynosi w(h)dh. Si&#x142;a r&#xF3;&#x17C;niczkowa dF dzia&#x142;aj&#x105;ca na ten pas wynosi:\begin{equation*}dF=P( \text{area})=(\rho gh)( w(h)dh)\end{equation*}&#xA0;Sumuj&#x105;c si&#x142;y dzia&#x142;aj&#x105;ce na wszystkie takie pasy w przedziale g&#x142;&#x119;boko&#x15B;ci od a do b, ca&#x142;kowita si&#x142;a hydrostatyczna F jest wyra&#x17C;ona jako ca&#x142;ka oznaczona:\begin{equation*}F=\int_a^b\rho gh w(h)dh\end{equation*}&#xA0;To r&#xF3;wnanie uwzgl&#x119;dnia zmiany szeroko&#x15B;ci &#x15B;ciany zapory wraz z g&#x142;&#x119;boko&#x15B;ci&#x105;, umo&#x17C;liwiaj&#x105;c dok&#x142;adne modelowanie zar&#xF3;wno powierzchni p&#x142;askich, jak i zakrzywionych.Metoda ta jest niezb&#x119;dna w takich dziedzinach in&#x17C;ynierii, jak in&#x17C;ynieria l&#x105;dowa i wodna. Zapewnia bezpieczne i efektywne projektowanie konstrukcji, takich jak zapory, mury oporowe, &#x15B;luzy i zbiorniki magazynowe, poprzez obliczenie wp&#x142;ywu zmiennego ci&#x15B;nienia cieczy na integralno&#x15B;&#x107; konstrukcji. Zdolno&#x15B;&#x107; ca&#x142;kowania do dok&#x142;adnego odwzorowania rozk&#x142;adu si&#x142; zapewnia in&#x17C;ynierom precyzj&#x119; niezb&#x119;dn&#x105; do zapewnienia stabilno&#x15B;ci i bezpiecze&#x144;stwa konstrukcji maj&#x105;cych kontakt z ciecz&#x105;.Rachunek r&#xF3;&#x17C;niczkowy i ca&#x142;kowy stanowi podstaw&#x119; nowoczesnych praktyk in&#x17C;ynieryjnych, przekszta&#x142;caj&#x105;c prawa fizyki w wyra&#x17C;enia ca&#x142;kowalne. Umo&#x17C;liwia analiz&#x119; i konstrukcj&#x119; z&#x142;o&#x17C;onych, rzeczywistych system&#xF3;w.

Zastosowania całkowania do wyznaczania ciśnienia hydrostatycznego

Si&#x142;a hydrostatyczna to ca&#x142;kowita si&#x142;a, z jak&#x105; ciecz w spoczynku dzia&#x142;a na powierzchni&#x119;. Dla poziomej powierzchni ...

applications-of-integration-to-find-hydrostatic-pressure

Education

JoVE Core

Calculus

Rachunek różniczkowy i całkowy

Unit

Jednostka

Application of Techniques of Integration

Zastosowanie technik całkowania

KEY TERMS AND CONCEPTS

KLUCZOWE TERMINY I POJĘCIA

hydrostatic force on dam walls with integration

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth.

A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall.

Integration helps measure this changing force. Pressure in a still fluid rises linearly with depth. At depth h, the pressure P depends on the fluid density ρ and gravity g.

To find the total force, the submerged surface is split into very thin horizontal strips. Each strip has width w(h) and small height dh, so its area is w(h)dh. The force on one strip is written as a differential force dF.

The forces on all strips are added with a definite integral from depth a to depth b. This method also works when the wall width changes with depth. It can model both flat and curved surfaces.

These calculations matter in civil and hydraulic engineering. Engineers use them to design dams, retaining walls, gates, and storage tanks safely. Integration gives the precision needed to handle changing fluid pressure and protect structural integrity.

Hydrostatic Force on Dam Walls with Integration

hydrostatic-force-on-dam-walls-with-integration

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applications of integration to find hydrostatic pressure

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth. A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall. Integration helps measure this changing force. Pressure in a still fluid rises...

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key terms and concepts

Learn how integration is used to calculate hydrostatic force on dam walls by adding the changing pressure of water across submerged surfaces.

Hydrostatic Force on Dam Walls with Integration | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Hydrostatic Force on Dam Walls with Integration

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