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

ينشأ الضغط الهيدروستاتيكي من وزن السائل في حالة السكون ويزداد بشكل متناسب مع العمق.بالنسبة لسطح أفقي مغمور بعمق ثابت، يكون الضغط ثابتا ويساوي حاصل ضرب كثافة السوائل، والتسارع الناتج عن الجاذبية، والعمق.لكن بالنسبة لسطح عمودي مثل وجه السد، كل نقطة تقع على عمق مختلف، مما يؤدي إلى زيادة الضغط مع العمق وخلق توزيع قوة غير منتظم.لحساب القوة الهيدروستاتيكية الكلية الناتجة عن هذا الضغط المتغير، يتم تقسيم المقطع المغمور إلى شرائط أفقية رفيعة. كل شريط يقع على عمق محدد وله مساحة تساوي عرضه مضروبا في سماكته الرأسية الصغيرة.الضغط على كل شريط يساوي كثافة وزن السائل مضروبا في عمقه تحت السطح، والقوة على ذلك الشريط تساوي الضغط مضروبا في المساحة.جمع هذه القوى عبر جميع الشرائط يقارب القوة الكلية باستخدام مجموع ريمان. مع اقتراب سمك الشريط من الصفر، يصبح المجموع تكاملا محددا.يعبر هذا التكامل عن القوة الهيدروستاتيكية الكلية كضغط مضروبا في دالة العرض، المدمجة على العمق المغمور.تعد هذه الطريقة ضرورية في الهندسة المدنية لتصميم هياكل احتجاز فعالة.

تطبيقات التكامل لإيجاد الضغط الهيدروستاتيكي

Applications of Integration to Find Hydrostatic Pressure

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

Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Application of Techniques of Integration

تطبيقات تقنيات التكامل

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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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Center of Mass in Discrete and Continuous Systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass. For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position,...

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Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

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Consumer Surplus from Demand and Integration

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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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