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

Pappus와 Guldinus는 모든 회전체의 표면적과 부피를 찾기 위해 정리를 개발했습니다.
표면적을 생성하려면 x축을 중심으로 알려진 길이의 평면 곡선을 회전합니다.
차동 선 요소를 고려하십시오. 회전하면 전체 표면적을 얻기 위해 통합된 고리가 생성됩니다. 
첫 번째 정리는 회전 표면의 면적이 생성 곡선의 길이와 곡선의 중심이 표면을 생성하는 동안 이동하는 거리의 곱이라고 말합니다.
마찬가지로, 회전 볼륨을 찾기 위해 평면 영역은 교차하지 않는 축을 중심으로 회전합니다.
회전할 때 체적 요소를 생성하는 차등 영역을 고려하십시오. 그런 다음 차동 부피가 통합되어 전체 부피를 결정합니다. 
두 번째 정리는 회전체의 부피는 생성 영역과 부피를 생성하는 동안 영역의 중심이 이동하는 거리의 곱이라고 말합니다.
두 경우 모두 회전이 각도를 통해서만 이루어지는 경우 공식이 그에 따라 변경됩니다.

<ol>
	<li>히벨러, R.C. (2016). 엔지니어링 역학: 정적. 14판, 뉴저지: 피어슨. 섹션 9.3, pp. 502-503.</li>
	<li>Meriam, J.L., Kraige, L.G. 및 Bolton, J.N. (2016). Engineering Mechanics: Statics and Dynamics(정역학 및 동역학). 여덟 번째 판, 싱가포르 : John Wiley & Sons. 섹션 5.5 pp. 261-262.</li>
</ol>

파푸스와 굴디누스의 정리

파푸스와 굴디누스가 개발한 두 가지 정리는 수학, 공학, 물리학에서 어떤 회전체의 표면적과 부피를 구하는 데 널리 사용됩니다. 이것은 평면 곡선을 교차하지 않는 축을 중심으로 회전시켜 표면적을 찾거나 평면 면적을 교차하지 않는 축을 중심으로 회전시켜 부피를 계산합니다.
표면적을 구하기 위해, 회전할 때 표면적 dA를 갖는 고리를 생성하는 미분선 요소를 생각해 보겠습니다. 이 미분적을 적분하면 회전의 표면적이 결정되고, 이것은 첫 번째 정리에 의해 기술됩니다. 첫 번째 정리는 회전의 표면적은 생성곡선의 길이와 표면적이 생성되는 동안 그 중심으로 이동한 거리의 곱과 같다는 것을 말합니다.
마찬가지로, 회전 부피를 계산하기 위해 회전할 때 미분 부피를 갖는 고리를 생성하는 미분적분을 고려해 보겠습니다. 두 번째 정리에 따라 이 미분적분을 적분하면 회전 부피가 결정됩니다. 파푸스와 굴디누스의 두 번째 정리는 회전 부피가 생성 면적과 그러한 부피를 생성할 때 중심으로 이동한 거리의 곱과 같다는 것을 말합니다. 이해할 만하게도 곡선이나 면적 중 어느 하나가 완전한 360° 회전이 아니라 각도를 통해 회전한다면 각각의 공식은 그에 따라 변경되어야 할 것입니다.

Pappus와 Guldinus가 개발한 두 가지 정리는 수학, 공학 및 물리학에서 모든 물체의 표면적과 부피를 찾는 데 널리 사용됩니다.

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Center of Gravity and Centroid

무게중심과 도심

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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...

centroid of a body

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...

centroid of a body: problem solving

Centroid of a Quarter Circle Wire

The centroid of a quarter circle wire shows how balance points can be found for curved shapes. In physics and engineering, the centroid helps describe stability, balance, and design. For a thin wire bent into a quarter circular arc, the goal is to find the centroid coordinates of the arc itself. Polar coordinates are used to solve this centroid problem. The wire is divided into small differential elements, and each element has a length equal to the radius times the differential angle. The x-...

centroid for the paraboloid of revolution

centroid-for-the-paraboloid-of-revolution

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...

resultant of a general distributed loading

resultant-of-a-general-distributed-loading

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...

fluid pressure

How Depth Changes Fluid Pressure

Fluid pressure changes with depth in liquids used in engineering systems. It matters in hydraulic systems, pumps, and valves, where the forces from moving and resting fluid must be considered to prevent damage or failure. For a fluid at rest, Pascal’s law says the pressure is equal in all directions. Pressure is force per unit area. Its size depends on the fluid’s specific weight, or mass density, and on how deep the measurement point is below the fluid surface. This relationship can be...

fluid pressure over flat plate of constant width

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

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

fluid-pressure-over-curved-plate-of-constant-width

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.

mechanical engineering

Surface Area and Volume by Centroid Paths

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