A uniform metal disck of radius R is taken and out of it a disc of diameter R is cut-off from the end. The center of mass of the remaining part will be

Options

A. `R/4` from the center
B. `R/3` from the center
C. `R/5` from the center
D. `R/6` from the center.

Explanation

To find the center of mass of the remaining part of a uniform metal disc after cutting out a smaller disc, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Geometry**:
- We have a uniform metal disc of radius \( R \).
- A smaller disc of diameter \( R \) (which means its radius is \( \frac{R}{2} \)) is cut out from the edge of the larger disc.

2. **Determine the Area of the Discs**:
- The area \( A_1 \) of the larger disc (radius \( R \)) is given by:
\[ A_1 = \pi R^2 \]
- The area \( A_2 \) of the smaller disc (radius \( \frac{R}{2} \)) is:
\[ A_2 = \pi \left(\frac{R}{2}\right)^2 = \pi \frac{R^2}{4} \]

3. **Calculate the Area of the Remaining Part**:
- The area of the remaining part \( A \) is:
\[ A = A_1 - A_2 = \pi R^2 - \pi \frac{R^2}{4} = \pi R^2 \left(1 - \frac{1}{4}\right) = \pi R^2 \cdot \frac{3}{4} = \frac{3\pi R^2}{4} \]

4. **Locate the Center of Mass of the Larger Disc**:
- The center of mass of the larger disc is at its center, which we can denote as point \( O \) (0, 0).

5. **Locate the Center of Mass of the Smaller Disc**:
- The center of mass of the smaller disc (cut-out) is at a distance \( R \) from the center of the larger disc along the radius, which we can denote as point \( C_2 \) (R, 0).

6. **Apply the Center of Mass Formula**:
- We will use the formula for the center of mass of the remaining part:
\[ A_1 x_1 - A_2 x_2 = A x \]
- Where:
- \( x_1 = 0 \) (center of the larger disc)
- \( x_2 = R \) (center of the smaller disc)
- \( A = \frac{3\pi R^2}{4} \) (area of the remaining part)
- Plugging in the values:
\[ \pi R^2 \cdot 0 - \pi \frac{R^2}{4} \cdot R = \frac{3\pi R^2}{4} \cdot x \]
- This simplifies to:
\[ -\frac{\pi R^3}{4} = \frac{3\pi R^2}{4} \cdot x \]

7. **Solve for \( x \)**:
- Dividing both sides by \( \frac{3\pi R^2}{4} \):
\[ x = -\frac{R}{3} \]

8. **Interpret the Result**:
- The negative sign indicates that the center of mass of the remaining part is located \( \frac{R}{3} \) units to the left of the center of the original disc.

### Final Answer:
The center of mass of the remaining part of the disc is located at a distance \( \frac{R}{3} \) from the center of the original disc towards the side from which the smaller disc was cut.

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