The distance of center of mass from the center of a uniform disc of radius R. if a circular plate is removed from the disc as shown in figure will be (C is centre of complete disc)

Options

A. `R/6`
B. `R/2`
C. `3R/8`
D. `R/4`

Answer

To solve the problem of finding the distance of the center of mass from the center of a uniform disc after a circular plate is removed, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components**:
- Let the radius of the original uniform disc be \( R \).
- When a circular plate of radius \( r \) is removed from the disc, we need to consider the center of mass of the remaining part of the disc.

2. **Define the Centers**:
- Let \( C \) be the center of the original disc.
- Let \( C_1 \) be the center of the removed circular plate.
- Let \( C_2 \) be the center of mass of the remaining portion of the disc.

3. **Mass Consideration**:
- Let \( m_1 \) be the mass of the removed circular plate.
- Let \( m_2 \) be the mass of the remaining part of the disc.

4. **Use the Center of Mass Formula**:
- The center of mass of the system can be calculated using the formula:
\[ m_1 \cdot \vec{C_1} + m_2 \cdot \vec{C_2} = 0 \]
- Rearranging gives:
\[ m_2 \cdot \vec{C_2} = -m_1 \cdot \vec{C_1} \]
- This indicates that the moment of the masses about the center of the original disc is zero.

5. **Express the Centers in Terms of Distances**:
- If we assume the distance from \( C \) to \( C_1 \) is \( R \) (the radius of the disc), then we can express \( \vec{C_1} \) as \( R \) in the direction away from \( C \).

6. **Calculate the Center of Mass of the Remaining Portion**:
- From the above equation, we can express \( \vec{C_2} \):
\[ \vec{C_2} = -\frac{m_1}{m_2} \cdot \vec{C_1} \]
- Since \( m_1 \) and \( m_2 \) are proportional to the areas of the discs, we can say:
\[ \frac{m_1}{m_2} = \frac{\text{Area of removed plate}}{\text{Area of remaining disc}} = \frac{\pi r^2}{\pi (R^2 - r^2)} = \frac{r^2}{R^2 - r^2} \]

7. **Substituting Values**:
- Substitute this ratio back into the equation for \( \vec{C_2} \):
\[ \vec{C_2} = -\frac{r^2}{R^2 - r^2} \cdot R \]

8. **Finding the Distance**:
- The distance from \( C \) to \( C_2 \) is thus:
\[ d = \frac{r^2}{R^2 - r^2} \cdot R \]
- For small values of \( r \) compared to \( R \), this simplifies to \( \frac{R}{2} \).

9. **Conclusion**:
- Therefore, the distance of the center of mass from the center of the uniform disc after removing the circular plate is \( \frac{R}{2} \).

### Final Answer:
The distance of the center of mass from the center of the uniform disc is \( \frac{R}{2} \).