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

Consider a cylindrical storage tank with a conical top.&#160;
The tank represents a composite body, which can be divided into a finite number of parts with simpler shapes.
Pappus and Guldinus&#39;s theorems can be applied to find the surface area and volume of such composite shapes.
To find the surface area, the generating curves for each shape are identified and revolved around a non-intersecting axis. Then, the centroid is located for each curve.
The length of the generating curve and the distance traveled by the curve&#39;s centroid are then used to obtain the total surface area of the tank.
Similarly, to estimate the volume, the generating area of each shape is identified and revolved around the same axis.
The volume of a body of revolution is equal to the product of the generating area and the distance traveled by the centroid of the shape.
By finding the area of each part and identifying their respective centroids, the values are substituted in the theorem to obtain the tank&#39;s volume.

theorems of pappus and guldinus: problem solving

Pappus and Guldinus&#39;s theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a cylinder and a cone.
To solve this problem, the first theorem is applied to find the surface area of the solid formed by revolving each of the generating curves around a non-intersecting axis. This involves substituting relevant values such as the length of the generating curve and the distance traveled by its centroid, obtaining separate values for each part, and then adding them up to obtain an overall total. For calculating the volume, we revolve an area around the same axis as before and use information from the second theorem, which involves multiplying generated areas by the distances traveled by their respective centroids.
By carrying out calculations with these two principles, it becomes easier to find accurate values for both the surface area and the volume of complex shapes such as this storage tank.

Theorems of Pappus and Guldinus: Problem Solving

Pappus and Guldinus&#39;s theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For ...

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

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Center of Gravity and Centroid

KEY TERMS AND CONCEPTS

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

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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...

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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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.

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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.

Pappus and Guldinus Theorems for Surface Area

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