Name: pappus and guldinus theorems for surface area
Uploaded: 2023-09-22T11:49:32.000Z
Duration: 1 min 12 s
Channel: JoVE

원뿔형 상단이 있는 원통형 저장 탱크를 고려하십시오. 
탱크는 더 단순한 모양을 가진 한정된 수의 부품으로 나눌 수 있는 복합 몸체를 나타냅니다.
Pappus와 Guldinus의 정리는 이러한 복합 모양의 표면적과 부피를 찾는 데 적용될 수 있습니다.
표면적을 찾기 위해 각 모양에 대한 생성 곡선을 식별하고 교차하지 않는 축을 중심으로 회전합니다. 그런 다음 중심이 각 곡선에 대해 배치됩니다.
생성 곡선의 길이와 곡선의 중심이 이동한 거리는 탱크의 총 표면적을 얻는 데 사용됩니다.
마찬가지로, 부피를 추정하기 위해 각 모양의 생성 영역이 식별되고 동일한 축을 중심으로 회전합니다.
회전체의 부피는 생성 면적의 곱과 모양의 중심이 이동한 거리와 같습니다.
각 부분의 면적을 찾고 각각의 중심을 식별함으로써 값은 탱크의 부피를 얻기 위해 정리에서 대체됩니다.

<ol>
	<li>히벨러, R.C. (2016). 엔지니어링 역학: 정적. 14판, 뉴저지: 피어슨. 섹션 9.3, 503, 507쪽.</li>
</ol>

파푸스와 굴디누스의 정리: 문제 해결

파푸스와 굴디누스의 정리는 복합 도형의 표면적과 부피를 구하는 데 사용되는 강력한 수학 원리입니다. 예를 들어, 원뿔 모양의 꼭대기를 가진 원통형 저장 탱크를 생각해 보세요. 이렇게 복잡한 모양의 경우 표면적이나 부피를 찾는 것이 어려울 수 있습니다. 이러한 정리는 부피와 표면적을 계산하는 데 특히 유용합니다. 여기서 원뿔 모양의 꼭대기를 가진 원통형 저장 탱크는 원뿔과 원뿔의 두 가지 간단한 모양으로 나눌 수 있습니다.
이 문제를 해결하기 위해 첫 번째 정리를 적용하여 각 생성 곡선을 교차하지 않는 축을 중심으로 회전하여 형성된 입체의 표면적을 찾습니다. 여기에는 생성 곡선의 길이와 중심으로 이동한 거리와 같은 관련 값을 대입하여 각 부분에 대해 별도의 값을 구한 다음 이를 합산하여 전체 합계를 구하는 것이 포함됩니다. 부피를 계산하기 위해 이전과 같은 축을 중심으로 면적을 회전하고 생성된 면적과 각 중심으로 이동한 거리를 곱한 두 번째 정리의 정보를 사용합니다.
이 두 가지 원리로 계산을 수행하면 이 저장 탱크와 같은 복잡한 형상의 표면적과 부피 모두에 대한 정확한 값을 찾는 것이 쉬워집니다.

Pappus와 Guldinus의 정리는 복합 모양의 표면적과 부피를 찾는 데 사용되는 강력한 수학적 원리입니다. 때문에...

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

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: 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 generating curve and the distance traveled by its centroid, or center of mass.

The surface area is found separately for each part and then added together. To find volume, the second theorem is used. It works by revolving an area around the same axis and multiplying the generated area by the distance traveled by its centroid. This makes it easier to calculate accurate surface area and volume for complex shapes like the storage tank.

Pappus and Guldinus Theorems for Surface Area

pappus-and-guldinus-theorems-for-surface-area

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

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...

theorems of pappus and guldinus: problem solving

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's theorems help calculate surface area and volume for composite shapes like a tank with a conical top.

Pappus and Guldinus Theorems for Surface Area | 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

Pappus and Guldinus Theorems for Surface Area

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