Find the centre of mass of a thin, uniform disc of radius R from which a small concentric disc of radius r is cut.

Answer

To find the center of mass of a thin uniform disc of radius \( R \) from which a small concentric disc of radius \( r \) is cut out, we can follow these steps:

### Step 1: Define the Discs and Their Masses
Let:
- The mass of the larger disc (radius \( R \)) be \( M \).
- The mass of the smaller disc (radius \( r \)) be \( m \).

### Step 2: Set Up the Coordinate System
Assume a coordinate system where the center of both discs is at the origin (0, 0). This is a symmetric arrangement.

### Step 3: Determine the Center of Mass of Each Disc
For a uniform disc, the center of mass is located at its geometric center. Therefore:
- The center of mass of the larger disc (Disc 1) is at \( (x_1, y_1) = (0, 0) \).
- The center of mass of the smaller disc (Disc 2) is also at \( (x_2, y_2) = (0, 0) \).

### Step 4: Apply the Center of Mass Formula
The center of mass \( (x_{cm}, y_{cm}) \) of the combined system can be calculated using the formula:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
\[ y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \]
Here, \( m_1 \) is the mass of the larger disc, and \( m_2 \) is the mass of the smaller disc (which we consider as negative since it is removed).

### Step 5: Substitute the Values
Substituting the values into the formulas:
- For the x-coordinate:
\[ x_{cm} = \frac{M \cdot 0 + (-m) \cdot 0}{M - m} = \frac{0}{M - m} = 0 \]
- For the y-coordinate:
\[ y_{cm} = \frac{M \cdot 0 + (-m) \cdot 0}{M - m} = \frac{0}{M - m} = 0 \]

### Step 6: Conclusion
Thus, the center of mass of the remaining shape after cutting out the smaller disc is:
\[ (x_{cm}, y_{cm}) = (0, 0) \]

### Final Answer
The center of mass of the remaining disc is at the origin \( (0, 0) \).

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