Center of Mass of Remaining Portion of the Disc

To determine the center of mass of the remaining portion of the disc, we need to consider the distribution of mass within the disc and the geometry of the removed portion. Let's break down the problem step by step.

1. Geometry of the Disc:
- The original disc has a uniform circular shape with a radius of R.
- The center of the disc is the point from which we will measure distances.

2. Removed Portion:
- A circular disc is removed from the original disc.
- The removed disc has a radius of R/6.
- The center of the removed disc is located at a distance of R/2 from the center of the original disc.

3. Remaining Portion:
- The remaining portion of the disc is obtained by subtracting the removed disc from the original disc.

4. Center of Mass:
- The center of mass of an object is the point where the entire mass of the object can be considered to be concentrated.
- In a uniform object, the center of mass is located at the geometric center or the centroid.

5. Center of Mass Calculation:
To find the center of mass of the remaining portion of the disc, we can consider the geometric properties of the disc.

- The center of the original disc is the origin (0, 0) in a Cartesian coordinate system.
- The center of the removed disc is located at a distance of R/2 from the origin in the positive x-direction.
- The remaining portion of the disc is symmetric about the y-axis.

6. Center of Mass Coordinates:
- The x-coordinate of the center of mass will be the same as the x-coordinate of the centroid of the remaining portion of the disc.
- The y-coordinate of the center of mass will be zero since the remaining portion is symmetric about the y-axis.

7. x-coordinate Calculation:
- The x-coordinate of the centroid of the remaining portion can be calculated using the formula for the centroid of a composite shape.
- Since the remaining portion consists of a full disc minus a smaller disc, we can consider it as a composite shape made up of two parts.
- The x-coordinate of the centroid of the remaining portion can be calculated as the weighted average of the x-coordinates of the centroids of the two parts.
- The weight of each part is proportional to its area.

8. Area Calculation:
- The area of a circle is given by the formula A = πr^2, where r is the radius.
- The area of the original disc is πR^2.
- The area of the removed disc is π(R/6)^2.

9. Centroid Calculation:
- The x-coordinate of the centroid of a full disc is zero since it is symmetric about the y-axis.
- The x-coordinate of the centroid of the removed disc can be calculated as the distance of its center from the y-axis, which is R/2.

10. Weighted Average Calculation:
- The x-coordinate of the center of mass is given by the weighted average of the x-coordinates of the centroids of the two parts.
- The weight of each part is proportional to its area.
- Using the weighted average formula, we can calculate the x