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

Hidrostatik basınç, hareketsiz haldeki sıvı ağırlığından kaynaklanır ve derinlikle orantılı olarak artar.Sabit bir derinlikte batırılan yatay bir yüzey için basınç sabittir ve sıvı yoğunluğu, yerçekiminden kaynaklanan ivme ve derinlik çarpımına eşittir.Ancak baraj yüzeyi gibi dikey bir yüzey için, her nokta farklı derinlikte bulunur, bu da basıncın derinlikle artmasına ve kuvvet dağılımı düzensiz olmasına neden olur.Bu değişken basınçtan kaynaklanan toplam hidrostatik kuvveti hesaplamak için, batan bölüm ince yatay şeritlere bölünür. Her şerit belirli bir derinlikte yer alır ve genişliğine eşit bir alana sahiptir, çarpımı küçük dikey kalınlıktır.Her şerit üzerindeki basınç, sıvının ağırlık yoğunluğunun yüzey altındaki derinliğinin çarpmasına eşittir ve o şerit üzerindeki kuvvet, basıncın alan ile çarpılmasına eşittir.Bu kuvvetlerin tüm şeritler üzerindeki toplamları, Riemann toplamı kullanılarak toplam kuvveti yaklaşık olarak hesaplar. Şerit kalınlığı sıfıra yaklaştıkça, toplam kesin bir integral olur.Bu integral, toplam hidrostatik kuvveti basınçla genişlik fonksiyonunun katı olarak ifade eder; bu fonksiyon sualtı derinlik üzerinde entegre edilir.Bu yöntem, verimli tutucu yapıların tasarımı için inşaat mühendisliğinde gereklidir.

hidrostatik kuvvetin bulunmasında İntegrasyonun uygulamaları

Applications of Integration to Find Hydrostatic Pressure

Hidrostatik kuvvet, bir ak&#x131;&#x15F;kan&#x131;n durgun durumda bir y&#xFC;zeye uygulad&#x131;&#x11F;&#x131; toplam kuvvettir. Sabit bir derinlikte ak&#x131;&#x15F;kana bat&#x131;r&#x131;lm&#x131;&#x15F; yatay bir y&#xFC;zey i&#xE7;in bas&#x131;n&#xE7; sabittir ve ak&#x131;&#x15F;kan&#x131;n yo&#x11F;unlu&#x11F;u, yer&#xE7;ekimi ivmesi ve derinli&#x11F;in &#xE7;arp&#x131;m&#x131; olarak hesaplan&#x131;r. Suya bat&#x131;r&#x131;lm&#x131;&#x15F; dikey bir baraj duvar&#x131; durumunda ise, derinlikle artan bas&#x131;n&#xE7; nedeniyle bu kuvvet y&#xFC;zey boyunca e&#x15F;it olarak da&#x11F;&#x131;lmaz. Bu de&#x11F;i&#x15F;im, her bir noktan&#x131;n &#xFC;zerindeki &#xFC;stteki suyun birikimli a&#x11F;&#x131;rl&#x131;&#x11F;&#x131;ndan kaynaklan&#x131;r. &#x130;ntegrasyon, toplam hidrostatik kuvveti do&#x11F;ru bi&#xE7;imde hesaplamak i&#xE7;in ak&#x131;&#x15F;kana bat&#x131;r&#x131;lm&#x131;&#x15F; y&#xFC;zey boyunca bas&#x131;n&#xE7;taki s&#xFC;rekli de&#x11F;i&#x15F;imi dikkate almak amac&#x131;yla kullan&#x131;l&#x131;r.Durgun bir ak&#x131;&#x15F;kanda bas&#x131;n&#xE7;, derinlikle do&#x11F;rusal olarak artar. h derinli&#x11F;inde bas&#x131;n&#xE7; P a&#x15F;a&#x11F;&#x131;daki gibi ifade edilir:\begin{equation*}P=\rho gh\end{equation*}&#xA0;burada &#x3C1; ak&#x131;&#x15F;kan&#x131;n yo&#x11F;unlu&#x11F;u, g ise yer&#xE7;ekimi ivmesidir.Toplam kuvveti hesaplamak i&#xE7;in, ak&#x131;&#x15F;kana bat&#x131;r&#x131;lm&#x131;&#x15F; y&#xFC;zey sonsuz ince yatay &#x15F;eritlere ayr&#x131;l&#x131;r. h derinli&#x11F;indeki her bir &#x15F;eridin geni&#x15F;li&#x11F;i w(h) ve diferansiyel y&#xFC;ksekli&#x11F;i dh oldu&#x11F;undan, alan&#x131; w(h)dh olur. Bu &#x15F;erit &#xFC;zerindeki diferansiyel kuvvet dF a&#x15F;a&#x11F;&#x131;daki gibidir:\begin{equation*}dF=P( \text{area})=(\rho gh)( w(h)dh)\end{equation*}&#xA0;a ile b aras&#x131;ndaki derinlik aral&#x131;&#x11F;&#x131; boyunca bu t&#xFC;r t&#xFC;m &#x15F;eritler &#xFC;zerindeki kuvvetler topland&#x131;&#x11F;&#x131;nda, toplam hidrostatik kuvvet F a&#x15F;a&#x11F;&#x131;daki belirli integral ile ifade edilir:\begin{equation*}F=\int_a^b\rho gh w(h)dh\end{equation*}&#xA0;Bu form&#xFC;lasyon, baraj duvar&#x131;n&#x131;n geni&#x15F;li&#x11F;inin derinlikle birlikte de&#x11F;i&#x15F;imini dikkate alarak hem d&#xFC;z hem de kavisli y&#xFC;zeylerin do&#x11F;ru bi&#xE7;imde modellenmesini sa&#x11F;lar.Bu y&#xF6;ntem, in&#x15F;aat m&#xFC;hendisli&#x11F;i ve hidrolik m&#xFC;hendisli&#x11F;i gibi alanlarda kritik &#xF6;neme sahiptir. De&#x11F;i&#x15F;ken ak&#x131;&#x15F;kan bas&#x131;nc&#x131;n&#x131;n yap&#x131;sal b&#xFC;t&#xFC;nl&#xFC;&#x11F;&#xFC; nas&#x131;l etkiledi&#x11F;ini hesaplayarak barajlar, istinat duvarlar&#x131;, kapaklar ve depolama tanklar&#x131; gibi yap&#x131;lar&#x131;n g&#xFC;venli ve etkili tasar&#x131;m&#x131;n&#x131; m&#xFC;mk&#xFC;n k&#x131;lar. &#x130;ntegrasyonun kuvvet da&#x11F;&#x131;l&#x131;m&#x131;n&#x131; tam olarak yakalama yetene&#x11F;i, m&#xFC;hendislere ak&#x131;&#x15F;kanla temas eden yap&#x131;larda kararl&#x131;l&#x131;k ve g&#xFC;venlik i&#xE7;in gereken hassasiyeti sa&#x11F;lar.Analiz, fiziksel yasalar&#x131; integre edilebilir ifadelere d&#xF6;n&#xFC;&#x15F;t&#xFC;rerek modern m&#xFC;hendislik uygulamalar&#x131;n&#x131;n temelini olu&#x15F;turur. Bu sayede karma&#x15F;&#x131;k, ger&#xE7;ek d&#xFC;nya sistemlerinin analiz edilmesi ve in&#x15F;a edilmesi m&#xFC;mk&#xFC;n olur.

Hidrostatik Kuvvetin Bulunmasında İntegrasyonun Uygulamaları

Hidrostatik kuvvet, bir ak&#x131;&#x15F;kan&#x131;n durgun durumda bir y&#xFC;zeye uygulad&#x131;&#x11F;&#x131; toplam kuvvettir. Sabit bir derinlikte ...

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

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Application of Techniques of Integration

İntegral Alma Tekniklerinin Uygulanması

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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

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calculus

Hydrostatic Force on Dam Walls with Integration

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