The center of mass for a uniform thin wire bent into a semicircle of radius r is determined using integration techniques. The correct formula for the y-coordinate of the center of mass is given by \(\bar y = \frac {\int_{-R}^{R} y \, dx}{\int_{-R}^{R} \, dx}\), where y is defined as \(\sqrt{r^2 - x^2}\). This approach emphasizes dividing by the diameter of the circle rather than half the area, ensuring accurate calculations. The discussion highlights the importance of proper integration limits and the relationship between units and the physical dimensions involved.
PREREQUISITESStudents and educators in physics, engineers working with structural designs, and anyone interested in the mathematical principles of mass distribution in physical objects.