Mechanical Engineering: Centroids & Center of Gravity (12 of 35) C. G. of a General Spandrel 2 By Michel van Biezen

Summary:

1. Introduction to Problem: The video focuses on finding the x-coordinate of the center of mass of a general spandrel. Due to space limitations, the x-coordinate is addressed separately from the y-coordinate, which was discussed in a previous video.

2. General Equation for Center of Mass: The general equation for calculating the x-coordinate of the center of mass of the spandrel is provided.

3. Definition of Small Segment $��$: In the horizontal direction, the coordinates of the center of mass of a small strip are determined. It's noted that the average distance from $�$ to $�+�$ is used.

4. Integral Setup and Limits: The integral for $�$ is set up, integrating from 0 to $�$ in the $�$ direction.

5. Substitution to Simplify the Integral: To simplify the integral, substitutions are made using equations for $�$ and ${�}^2$ in terms of $�$.

6. Integration and Simplification: The integral is evaluated, simplifying along the way and factoring out constants.

7. Further Simplification and Factoring: The resulting expression is simplified further, factoring out common terms.

8. Common Denominator and Simplification: After finding common denominators and simplifying, the equation is reduced to a simpler form.

9. Final Result: The x-coordinate of the center of mass of the spandrel is determined to be $\frac{�\cdot (�+1)}{�+2}$.

10. Conclusion: The video concludes by highlighting that despite the complexity, it's primarily algebraic manipulation once the integral is solved. This result will be useful for finding the center of mass of various odd-shaped objects in future examples.