Mechanical Engineering: Centroids & Center of Gravity (33 of 35) Volume=? using Pappus-Guldinus By Michel van Biezen
Summary:
Introduction and Objective: The video introduces the concept of finding the volume of a doughnut-shaped object formed by revolving a circular area around the x-axis. The objective is to apply the theorem of Pappus-Guldinus to calculate the volume.
Theorem Explanation: The Pappus-Guldinus theorem states that the volume of the object can be found by multiplying the area of the face being revolved around the x-axis by the distance traveled by the centroid of that area as it revolves. The distance traveled forms a circular path, and its length is 2Ï€ times the radius of the circular path.
Calculations with Numerical Values: Given that the distance from the axis of rotation to the centroid of the circular area is 10 centimeters, the radius of the circular path is determined to be 10 centimeters. Substituting values into the formula, the volume is calculated to be 1776 cubic centimeters.
Generalized Formula: The process is then described in terms of symbols, where R represents the radius of the circular path and Y with a line over it represents the distance from the x-axis to the centroid. The volume formula is given as Volume = πR^2 * 2πY.
Conclusion: The video concludes by presenting both the calculated numerical value and the general formula for finding the volume of a doughnut-shaped object using the theorem of Pappus-Guldinus. It emphasizes the simplicity of the method once the appropriate parameters are identified.
In summary, the video demonstrates how to find the volume of a doughnut-shaped object formed by revolving a circular area around the x-axis using the theorem of Pappus-Guldinus. It provides both numerical calculations and a generalized formula for the process.