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

考虑一个带有圆锥形顶部的圆柱形储罐。&#160;
该容器代表一个复合体，可被划分为若干个形状较简单的部分。
可以应用帕普斯和古尔丁定理来求解此类复合形状的表面积和体积。
为了求得表面积，需先确定每种形状的母线，并将其绕一个不相交的轴旋转。然后，确定每条曲线的形心位置。
然后利用生成曲线的长度和曲线形心移动的距离，计算出储罐的总表面积。
类似地，为了估算体积，需确定每种形状的生成面积，并将其绕同一轴旋转。
旋转体的体积等于生成面积与形状形心所经过的距离的乘积。
通过计算各部分的面积并确定其各自的形心，将这些值代入定理中，即可求得储罐的体积。

pappus和guldinus定理：问题求解

Pappus和Guldinus定理是强大的数学原理，能够用来计算复合形状的表面积和体积。例如，假设有一个带有锥形顶部的圆柱形储罐。对于这种复杂的形状，需要找到其相应的表面积或体积，并且这可能是具有挑战性的。这些定理在计算此类系统的体积和表面积时是特别有用的。在这里，带有锥形顶部的圆柱形储罐可以分解为两种简单的形状：圆柱体和圆锥体。 为了能够解决这个问题，首先使用第一个定理来找到由每个生成曲线绕非相交轴旋转形成的固体的表面积。这涉及到替换相关的值，如生成曲线的长度和其质心所移动的距离，以此来得出位于每个部分单独的值，然后将它们进行相加以得出总和。为了计算其中的体积，我们可以绕之前相同的轴来旋转一个区域，并使用第二个定理所提供的信息，该定理涉及到将其中生成的区域乘以其相应的质心所移动的距离。 通过使用这两个原理来进行计算，便可以更容易地找得出复杂形状（如这个储罐）的准确表面积和体积值。

Pappus和Guldinus定理：问题求解

帕普斯和古尔丁斯定理是用于求解复合形状表面积和体积的有力数学原理。对于……

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

Education

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Unit 1

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Center of Gravity and Centroid

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KEY TERMS AND CONCEPTS

关键术语与概念

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

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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...

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composite bodies

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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

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fluid pressure

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

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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.

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Pappus and Guldinus Theorems for Surface Area

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