Mechanical Engineering: Centroids & Center of Gravity (27 of 35) Area=? using Pappus-Guldinus By Michel van Biezen

Summary:

1. Introduction: The video introduces the second example, which involves a more challenging problem.

2. Problem Description: A circular curve, a quarter of a circle, is rotated about the x-axis, forming a semi-sphere or half a sphere.

3. Surface Area Calculation: The theorem of Pappus Guldinus states that the area is equal to the length of the curve multiplied by the distance covered by the centroid as it rotates about the x-axis.

4. Location of Centroid: The centroid of a quarter circle lies along a 45-degree line, with the y-coordinate equal to twice the radius of the circle.

5. Circular Path of Centroid: The path of the centroid forms a circle, with a length of $2�$ times the radius (in this case, $2�\mathrm{/}�$).

6. Application of Values: Assuming a radius of 5 centimeters, the distance covered by the centroid is $2�$ times $2�\mathrm{/}�$.

7. Calculation of Curve Length: The length of a quarter circle is $1\mathrm{/}4$ of the length of a full circle, which is $2�$ times the radius. This is multiplied by $2�$ and $2�\mathrm{/}�$.

8. Simplification: After simplifying the expression, it equals $2�{�}^2$, which makes sense as it's half the area of a full circle.

9. Verification of Result: The area calculated matches the expected area of half a sphere, confirming the validity of the theorem of Pappus Guldinus.

10. Conclusion: The theorem proves to be straightforward and effective, where the length of the curve is multiplied by the path length of the centroid to determine the area.

In summary, the video demonstrates the application of the theorem of Pappus Guldinus to find the area of a semi-sphere formed by rotating a quarter circle about the x-axis. The process involves calculating the path length of the centroid and applying it to the area formula, confirming the validity of the theorem.