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

パップスとグルディヌスは、あらゆる回転体の表面積と体積を求める定理を開発した。
サーフェス領域を生成するには、既知の長さの平面カーブを x 軸を中心に回転させます。
差動ライン要素について考えてみます。回転するとリングが生成され、リングが統合されて表面積全体が得られます。 
最初の定理は、回転面の面積は、生成曲線の長さと、曲面の生成中に曲線の重心が移動する距離の積であると述べています。
同様に、回転量を見つけるために、平面領域は交差しない軸を中心に回転します。
差動領域を考え、回転したときに体積要素を生成します。次に、差動ボリュームが積分されて、全体のボリュームが決定されます。 
2 番目の定理は、回転体の体積は、生成領域と、体積を生成するときに領域の重心が移動する距離の積であると述べています。
どちらの場合も、回転が角度のみの場合、式はそれに応じて変化します。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス:静力学。第14版、ニュージャージー州:ピアソン。セクション9.3、502-503ページ。</li>
	<li>メリアム、JL、クレイジ、LGおよびボルトン、JN(2016)。エンジニアリングメカニクス:静力学とダイナミクス。第8版、シンガポール:John Wiley & Sons。セクション5.5、261-262ページ。</li>
</ol>

パップスとギュルダンの定理

パップスとギュルダンによって開発された2つの定理は、回転体の表面積と体積を求めるために数学、工学、物理学で広く使用されています。これは、曲線を回転軸に沿って回転させてその表面積を求めたり、平面領域を交差しない軸周りに回転させて体積を計算することで行われます。

表面積を求めるためには、回転時に微小な線要素が生成するリングの表面積dAを考慮します。この微小な面積を積分することで回転体の表面積を求めることができます。この定理によれば、表面積は生成曲線の長さとその重心が表面積生成中に移動する距離の積と等しいです。

同様に、回転体の体積を計算するためには、微小な面積要素が回転時に生成する微小な体積リングを考慮します。この微小な体積を積分することで回転体の体積を求めることができます。パップスとギュルダンの第2の定理によれば、回転体の体積は生成領域とその重心が体積生成中に移動する距離の積と等しいです。理解できるように、曲線または領域が360°回転する代わりに角度で回転する場合、それぞれの式を適宜変更する必要があります。

PappusとGuldinusによって開発された2つの定理は、数学、工学、物理学で広く使用されており、あらゆる物体の表面積と体積を求めています...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Center of Gravity and Centroid

重心と図心

KEY TERMS AND CONCEPTS

主要用語と概念

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

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Surface Area and Volume by Centroid Paths

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