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

La pressione idrostatica deriva dal peso del fluido a riposo e aumenta proporzionalmente alla profondità.Per una superficie orizzontale sommersa a una profondità fissa, la pressione è costante e corrisponde al prodotto della densità del fluido, dell'accelerazione dovuta alla gravità e della profondità.Ma per una superficie verticale come la faccia di una diga, ogni punto si trova a una profondità diversa, causando un aumento della pressione con la profondità e creando una distribuzione delle forze non uniforme.Per calcolare la forza idrostatica totale da questa pressione variabile, la sezione sommersa viene suddivisa in sottili strisce orizzontali. Ogni striscia si trova a una profondità specifica e ha un'area pari alla sua larghezza moltiplicata per un piccolo spessore verticale.La pressione su ciascuna striscia è uguale alla densità di peso del fluido moltipitata per la sua profondità sotto la superficie, e la forza su quella striscia è uguale alla pressione moltiplicata per l'area.La somma di queste forze su tutte le strisce approssima la forza totale usando una somma di Riemann. Quando lo spessore della striscia si avvicina a zero, la somma diventa un integrale definito.Questo integrale esprime la forza idrostatica totale come pressione moltiplicata per la funzione di larghezza, integrata sulla profondità sommersa.Questo metodo è essenziale nell'ingegneria civile per progettare strutture di contenimento efficienti.

applicazioni dell’integrazione per determinare la pressione idrostatica

Applications of Integration to Find Hydrostatic Pressure

La forza idrostatica &#xE8; la forza totale esercitata da un fluido in quiete su una superficie. Per una superficie orizzontale immersa a profondit&#xE0; costante, la pressione &#xE8; costante e si calcola come il prodotto della densit&#xE0; del fluido, dell&#x2019;accelerazione di gravit&#xE0; e della profondit&#xE0;. Nel caso di un paramento verticale di una diga immerso in acqua, invece, la forza non &#xE8; distribuita in modo uniforme, poich&#xE9; la pressione aumenta con la profondit&#xE0;. Tale variazione deriva dal peso complessivo della colonna d&#x2019;acqua sovrastante ciascun punto. L&#x2019;integrazione consente di tenere conto della variazione continua della pressione lungo la superficie immersa, cos&#xEC; da determinare con precisione la forza idrostatica totale.La pressione in un fluido in quiete aumenta linearmente con la profondit&#xE0;. Alla profondit&#xE0; h, la pressione P &#xE8; data da:\begin{equation*}P=\rho gh\end{equation*}&#xA0;dove &#x3C1; &#xE8; la densit&#xE0; del fluido e g &#xE8; l&#x2019;accelerazione di gravit&#xE0;.Per calcolare la forza totale, la superficie immersa viene suddivisa in strisce orizzontali infinitesimali. Ogni striscia alla profondit&#xE0; h ha larghezza w(h) e spessore differenziale dh, quindi la sua area &#xE8; w(h)dh. La forza differenziale dF su quella striscia &#xE8;:\begin{equation*}dF=P( \text{area})=(\rho gh)( w(h)dh)\end{equation*}&#xA0;Sommando le forze su tutte queste strisce nell&#x2019;intervallo di profondit&#xE0; da a a b, la forza idrostatica totale F si esprime come l&#x2019;integrale definito:\begin{equation*}F=\int_a^b\rho gh w(h)dh\end{equation*}&#xA0;Questa formulazione tiene conto delle variazioni della larghezza del paramento della diga al variare della profondit&#xE0;, consentendo una modellazione accurata sia di superfici piane sia di superfici curve.Questo metodo &#xE8; essenziale in ambiti quali l&#x2019;ingegneria civile e l&#x2019;ingegneria idraulica. Esso garantisce la progettazione sicura ed efficace di strutture come dighe, muri di contenimento, paratoie e serbatoi di stoccaggio, permettendo di determinare in che modo la pressione variabile del fluido incida sull&#x2019;integrit&#xE0; strutturale. La capacit&#xE0; dell&#x2019;integrazione di rappresentare con esattezza la distribuzione della forza fornisce agli ingegneri la precisione necessaria per assicurare stabilit&#xE0; e sicurezza alle strutture a contatto con i fluidi.Il calcolo infinitesimale &#xE8; alla base delle moderne pratiche ingegneristiche, poich&#xE9; trasforma le leggi fisiche in espressioni integrabili. Ci&#xF2; consente l&#x2019;analisi e la realizzazione di sistemi complessi del mondo reale.

Applicazioni dell’integrazione per determinare la pressione idrostatica

La forza idrostatica &#xE8; la forza totale esercitata da un fluido in quiete su una superficie. Per una superficie orizzontale immersa a profondit&#xE0; ...

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

Education

Didattica

JoVE Core

Calculus

Analisi matematica

Unit

Unità

Application of Techniques of Integration

Applicazione delle tecniche di integrazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

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.

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calculus

Hydrostatic Force on Dam Walls with Integration

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