Name: surface area and volume by centroid paths
Uploaded: 2023-09-22T11:49:32.000Z
Duration: 1 min 10 s
Channel: JoVE

Папп и Гульдин разработали теоремы, чтобы найти площадь поверхности и объем любого тела вращения.
Чтобы получить площадь поверхности, поверните плоскую кривую известной длины вокруг оси x.
Рассмотрим элемент дифференциальной линии. При вращении он генерирует кольцо, которое интегрируется для получения всей площади поверхности. 
Первая теорема гласит, что площадь поверхности вращения является произведением длины порождающей кривой на расстояние, которое проходит центроид кривой при создании поверхности.
Аналогично, чтобы найти объем вращения, плоская область вращается вокруг непересекающейся оси.
Рассмотрим дифференциальную область, генерирующую объемный элемент при вращении. Затем дифференциальный объем интегрируется для определения всего объема. 
Вторая теорема гласит, что объем тела вращения является произведением порождающей площади и расстояния, которое проходит центроид этой области при создании объема.
В обоих случаях формула изменяется соответствующим образом, если вращение происходит только через угол.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика: Статика. Четырнадцатое издание, Нью-Джерси: Пирсон. Раздел 9.3, стр. 502-503.</li>
	<li>Мериам, Д.Л., Крейдж, Л.Г. и Болтон, Д.Н. (2016). Инженерная механика: статика и динамика. Восьмое издание, Сингапур: John Wiley & Sons. Раздел 5.5 С. 261-262.</li>
</ol>

Теоремы Паппуса и Гульдина

Две теоремы разработанные Паппусом и Гульдиным широко используются в математике, инженерии и физике для нахождения площади поверхности и объема любого вращающегося тела. Это делается путем вращения плоской кривой вокруг оси, которая не пересекает кривую, для нахождения площади поверхности или вращения плоской области вокруг непересекающейся оси для расчета ее объема.
Для нахождения площади поверхности рассмотрим дифференциальный элемент линии, который генерирует кольцо с площадью dA при вращении. Интегрирование этой дифференциальной площади определяет площадь поверхности вращения, которая описывается первой теоремой. Первая теорема утверждает, что площадь поверхности вращения равна произведению длины генерирующей кривой и расстояния, пройденного ее центроидом в процессе создания площади поверхности.
Аналогично, для расчета объема вращения рассмотрим дифференциальный элемент площади, который генерирует кольцо с дифференциальным объемом при вращении. Интегрирование этого дифференциального объема определяет объем вращения в соответствии со второй теоремой. Вторая теорема Паппуса и Гульдина утверждает, что объем вращения равен произведению генерирующей площади и расстояния, пройденного ее центроидом в процессе создания такого объема. Естественно, если какая-либо кривая или область вращается на угол, а не на полный оборот 360°, их соответствующие формулы должны быть изменены соответственно.

Две теоремы, разработанные Паппусом и Гульдином, широко используются в математике, технике и физике для определения площади поверхности и объема любого тела.

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Center of Gravity and Centroid

Центр тяжести и центр масс (центроид)

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

Theorems of Pappus and Guldinus

surface area and volume by centroid paths

Pappus and Guldinus theorems help find the surface area and volume of bodies of revolution. A body of revolution is formed when a plane curve or plane area is turned around an axis that does not intersect it.

For surface area, a curve is revolved around a non-intersecting axis. A small line element on the curve creates a thin ring as it moves. Adding these ring areas gives the total surface area of the solid.

The first theorem says that this surface area equals the length of the generating curve multiplied by the distance traveled by its centroid. The centroid is the geometric center of the curve as it moves.

For volume, a plane area is revolved around a non-intersecting axis. A small area element creates a ring with a tiny volume as it turns. Integrating these small volumes gives the total volume of revolution.

The second theorem says that this volume equals the generating area multiplied by the distance traveled by its centroid. If the curve or area turns through an angle instead of a full 360° rotation, the formula must be adjusted. These ideas are useful in mathematics, engineering, and physics.

Surface Area and Volume by Centroid Paths

surface-area-and-volume-by-centroid-paths

center of gravity

Center of Gravity in Balance and Design

The center of gravity is the point where an object’s weight appears to act. It is the spot used to balance the object most evenly. In mechanics, this point helps explain stability and moments of inertia. The center of gravity does not always sit inside the object’s shape. In some cases, it can lie outside the body. That is why the idea is useful for both simple and more complex shapes. Its location can be found with the principle of moments. The object is divided into very small parts, and...

center of mass

Center of Mass in Motion

The center of mass helps explain how an object moves through space. It is the point where the total mass of an object can be treated as concentrated. This idea applies to objects of different shapes and sizes. The center of mass is related to the center of gravity, but they are not the same. The center of gravity is the point where an object’s weight seems to be concentrated. It can be used to balance an object perfectly. Knowing the center of mass also helps predict motion. When a ball is...

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Finding the Center of a Body

The centroid is the geometric center of a body. It is an important idea in engineering, physics, and mechanics. For many objects, it lies inside the body. A body with holes or cavities can be an exception. For a homogeneous body, the centroid matches the center of mass or the center of gravity. Homogeneous means the material has a uniform density. In that case, the centroid can often be found by balancing the moments of the body’s volume. Symmetry can also help locate the centroid. If a...

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Centroid of a Quarter Circle Wire

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Centroid and Uses of a Paraboloid Shape

The centroid of a paraboloid of revolution is the point where its mass is balanced. A paraboloid of revolution is an axially symmetric surface made by rotating a parabola around its axis. In engineering, this location matters because it helps show how forces spread across the shape. Knowing the centroid can help when building parts that must handle stress and pressure reliably. The paraboloid shape has useful structural properties, including rotational symmetry and strength against stress...

composite bodies

Center of Gravity in Composite Bodies

Composite bodies are made of multiple connected parts that act as one larger object. Each part still has its own weight and center of gravity. To find the center of gravity of the whole body, engineers must account for every part. When the density or specific weight is constant, the center of gravity matches the centroid. The centroid is the geometric center of a shape. This idea is useful when analyzing parts that behave as a single unit. Composite bodies are common in mechanical engineering.

theorems of pappus and guldinus

Pappus and Guldinus theorems help find the surface area and volume of bodies of revolution. A body of revolution is formed when a plane curve or plane area is turned around an axis that does not intersect it. For surface area, a curve is revolved around a non-intersecting axis. A small line element on the curve creates a thin ring as it moves. Adding these ring areas gives the total surface area of the solid. The first theorem says that this surface area equals the length of the generating...

theorems of pappus and guldinus: problem solving

theorems-of-pappus-and-guldinus-problem-solving

Pappus and Guldinus Theorems for Surface Area

Pappus and Guldinus's theorems help find the surface area and volume of composite shapes. These rules are useful when a solid is made from simple parts that are hard to measure directly. A cylindrical storage tank with a conical top is one example. The tank can be split into two simple shapes: a cylinder and a cone. To find surface area, the first theorem is used for the solid formed by revolving each generating curve around a non-intersecting axis. The calculation uses the length of the...

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Finding the Resultant of Distributed Loads

Distributed loads create a force that is spread over a surface instead of acting at one point. To analyze this kind of loading, engineers use a single resultant force. That force represents the net effect of the entire load. This idea is important in structures that face non-uniform loads. Wind can push on buildings and towers with changing speed and direction, so the load is not evenly spread. Bridges also experience non-uniform loading from vehicles, and some parts of the bridge can feel...

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How Depth Changes Fluid Pressure

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fluid pressure over flat plate of constant width

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Center of Pressure on a Submerged Plate

Fluid pressure on a submerged flat plate depends on depth and shape. A plate under water experiences pressure acting normal to its surface, and the force is spread across the whole area. For a rectangular plate of constant width, this pressure increases from the top edge to the bottom edge. Because the pressure rises with depth, the loading forms a trapezoidal distribution. The resultant force on the plate comes from the trapezoidal volume of water with a specific weight. The line of action of...

fluid pressure over curved plate of constant width

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Finding Force on a Curved Plate in Water

Fluid pressure on a curved plate of constant width changes in both size and direction. That makes the total force hard to find directly. A common method is to split the force into horizontal and vertical parts and then combine them. The calculation becomes simpler by replacing the distributed load with an equivalent two-dimensional loading. The first step is to include the weight of the liquid inside the block. That force equals the specific weight of the liquid times the plate thickness and...

fluid pressure over flat plate of variable width

fluid-pressure-over-flat-plate-of-variable-width

A submerged flat plate experiences fluid pressure that changes with depth. This pressure can create effects such as drag and buoyancy. To study a flat plate with variable width, the pressure distribution across the plate must be analyzed first. The force on a small strip of the plate is found by multiplying the pressure by the strip’s differential area. The pressure can also be written using the fluid’s specific weight and the depth of that strip below the surface. To find the total, or...

key terms and concepts

Learn how Pappus and Guldinus theorems use centroid motion to find surface area and volume of bodies of revolution, with 360° and angled rotations.

Surface Area and Volume by Centroid Paths | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

Surface Area and Volume by Centroid Paths

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