Find the centre of mass of `a` uniform disc of radius a from which a circualr section of radius `b` has been removed. The centre of hole is at a distance `c` from the centre of the disc.

Options

A. `(pib^(2))/((a^(2)-b^(2)))`
B. `(-cb^(2))/((a^(2)-b^(2)))`
C. `(pc^(2))/((a^(2)-b^(2)))`
D. `(pia^(2))/((c^(2)-b^(2)))`

Explanation

To find the center of mass of a uniform disc of radius \( A \) from which a circular section of radius \( B \) has been removed, with the center of the hole at a distance \( C \) from the center of the disc, we can follow these steps:

### Step 1: Define the Masses
Let:
- \( M_1 \) be the mass of the original disc.
- \( M_2 \) be the mass of the removed circular section.

The mass of the original disc can be calculated using the area and the density (\( \sigma \)):
\[ M_1 = \pi A^2 \sigma \]
The mass of the removed section is:
\[ M_2 = \pi B^2 \sigma \]

### Step 2: Determine the Coordinates
- The center of the original disc is at the origin (0, 0).
- The center of the removed section is at the coordinates \( (C, 0) \).

### Step 3: Calculate the Center of Mass of the Remaining Structure
The center of mass \( x_{cm} \) of the remaining structure after removing the circular section can be calculated using the formula:
\[ x_{cm} = \frac{M_1 \cdot x_1 - M_2 \cdot x_2}{M_1 - M_2} \]
where:
- \( x_1 = 0 \) (the center of the original disc)
- \( x_2 = C \) (the center of the removed section)

Substituting the values:
\[ x_{cm} = \frac{M_1 \cdot 0 - M_2 \cdot C}{M_1 - M_2} \]
\[ x_{cm} = \frac{-M_2 \cdot C}{M_1 - M_2} \]

### Step 4: Substitute the Masses
Now substitute the expressions for \( M_1 \) and \( M_2 \):
\[ x_{cm} = \frac{-\pi B^2 \sigma \cdot C}{\pi A^2 \sigma - \pi B^2 \sigma} \]
The \( \pi \) and \( \sigma \) cancel out:
\[ x_{cm} = \frac{-B^2 C}{A^2 - B^2} \]

### Step 5: Interpret the Result
Since \( x_{cm} \) is negative, it indicates that the center of mass of the remaining structure is located to the left of the origin.

### Final Result
The center of mass of the remaining structure is:
\[ x_{cm} = -\frac{B^2 C}{A^2 - B^2} \]