Understanding the Problem
To find the maximum distance of the center of mass of the remaining mass after cutting a sphere of diameter r from a sphere of radius r, we need to analyze the geometry and mass distribution.
Concept of Center of Mass
- The center of mass of a solid sphere is located at its geometric center.
- When a smaller sphere is removed, the center of mass of the remaining material shifts.
Geometric Setup
- The larger sphere has a radius (R) of r.
- The smaller sphere cut from it has a diameter of r, hence a radius (r_s) of r/2.
Calculating the Shift in Center of Mass
- The center of the larger sphere is at the origin (0, 0, 0).
- The center of the smaller sphere can be placed at (0, 0, h), where h is the distance from the center of the larger sphere.
- To maximize the distance of the center of mass of the remaining mass, h should be set to r/2.
Final Calculation
- The distance from the original center (0, 0, 0) to the center of mass of the remaining mass can be calculated using the formula for center of mass after removing the smaller sphere.
- The effective distance is maximized when the smaller sphere is positioned such that its center is at the edge of the larger sphere, leading to a distance of r/2.
Conclusion
- The maximum distance of the center of mass of the remaining mass from the original center is r/2.
- This configuration ensures that the remaining mass is at its farthest distance, taking into consideration the symmetry and mass distribution of both spheres.