Problem:
A circular disc of radius R is removed from a bigger disc of radius 2R from one edge of the disc. Find the position of the center of mass of the residue disc.

Solution:

1. Calculating the Area:
To find the position of the center of mass of the residue disc, we first need to calculate the area of the residue disc.

The area of a circle is given by the formula A = πr^2, where r is the radius of the circle.

The area of the bigger disc is A1 = π(2R)^2 = 4πR^2.

The area of the smaller disc that is removed is A2 = πR^2.

Therefore, the area of the residue disc is A_residue = A1 - A2 = 4πR^2 - πR^2 = 3πR^2.

2. Calculating the Distance:
Next, we need to calculate the distance from the center of mass of the residue disc to the center of the bigger disc.

Since the residue disc is formed by removing a smaller disc from the bigger disc, the center of mass of the residue disc will lie on the line connecting the centers of the two discs.

Let's assume the distance from the center of the bigger disc to the center of mass of the residue disc is 'x'.

The distance from the center of the bigger disc to the edge of the residue disc is R, and the distance from the center of mass of the residue disc to the edge of the residue disc is R/2 (since the radius of the residue disc is R/2).

Therefore, the total distance from the center of the bigger disc to the center of mass of the residue disc is R + R/2 = 3R/2.

So, x = 3R/2.

3. Conclusion:
The position of the center of mass of the residue disc is located at a distance of 3R/2 from the center of the bigger disc, along the line connecting the centers of the two discs.

Its just a simple formula:
Shift = area removed × distance btw them/ Initial area - area removed