9.8:
Theorems of Pappus and Guldinus: Problem Solving
Consider a cylindrical storage tank with a conical top.
The tank represents a composite body, which can be divided into a finite number of parts with simpler shapes.
Pappus and Guldinus'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'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's volume.
9.8:
Theorems of Pappus and Guldinus: Problem Solving
Pappus and Guldinus'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.
Önerilen Okuma
- Hibbeler, R.C. (2016). Engineering Mechanics: Statics. Fourteenth Edition, New Jersey: Pearson. Section 9.3, Pp. 503, 507.