Mechanical Engineering: Centroids & Center of Gravity (30 of 35) Area, Vol=? using Pappus-Guldinus By Michel van Biezen

Summary:

1. Introduction: The video introduces two theorems of Pappus Guldinus side by side. One involves rotating a line segment, and the other involves rotating an area about an axis. The objective is to find the surface area and volume of the resulting shapes.

2. Rotating a Line Segment: When a semicircle line segment is rotated about the x-axis, it forms a sphere. The theorem states that the surface area of the sphere can be found by multiplying the length of the line segment by the distance traveled by its centroid as it rotates. The path followed by the centroid is a circle, so the distance is 2π times the radius of the circle. The radius is the y-coordinate of the centroid, which is 2 times the radius of the semicircle divided by π.

3. Rotating an Area: Similarly, when a semicircular area is rotated about the x-axis, it also forms a sphere. The theorem states that the volume of the sphere can be found by multiplying the surface area of the semicircle by the distance traveled by its centroid. The distance traveled is again in a circular path, but this time the radius is 4 times the radius of the semicircle divided by 3π.

4. Calculation for Surface Area of Sphere: For the line segment, the length of the semicircle is $2��$, and when squared, it yields $4�{�}^2$, which is indeed the surface area of a sphere.

5. Calculation for Volume of Sphere: For the area, the surface area of the semicircle is $\frac{1}{2}�{�}^2$, which, when multiplied by $2�$ and $4�\mathrm{/}3�$, results in $\frac{4}{3}�{�}^3$, which is the volume formula for a sphere.

6. Conclusion: The video concludes by highlighting the effectiveness of the Pappus Guldinus theorem in calculating surface areas and volumes of complex shapes. It emphasizes that while the theorem may not be necessary for spheres, it becomes invaluable for more complex shapes where surface area and volume calculations are challenging.

In summary, the video demonstrates the application of the Pappus Guldinus theorem to find the surface area and volume of spheres formed by rotating line segments and areas about the x-axis. It underscores the theorem's utility in simplifying calculations for various shapes.