A circular disc of radius R is removed from a bigger circular disc of radius 2R such that the cirucmferences of the discs coincide. The centre of mass of the new disc is `alpha/R` from the center of the bigger disc. The value of `alpha` is

Options

A. `1/4`
B. `1/3`
C. `1/2`
D. `1/6`

Explanation

To solve the problem of finding the value of \(\alpha\) for the center of mass of the new disc formed after removing a smaller disc from a larger disc, we can follow these steps:

### Step 1: Define the Masses
Let the mass per unit area (surface density) of the discs be \(\sigma\).

- The mass of the larger disc (radius \(2R\)) is given by:
\[ m_1 = \sigma \cdot \text{Area of larger disc} = \sigma \cdot \pi (2R)^2 = 4\pi R^2 \sigma \]

- The mass of the smaller disc (radius \(R\)) is given by:
\[ m_2 = \sigma \cdot \text{Area of smaller disc} = \sigma \cdot \pi R^2 = \pi R^2 \sigma \]

### Step 2: Determine the Center of Mass Positions
- The center of mass of the larger disc is at the origin (0,0).
- The center of mass of the smaller disc is at a distance \(R\) from the origin (since it is removed from the edge of the larger disc).

### Step 3: Apply the Center of Mass Formula
The center of mass \(x_{cm}\) of the remaining shape can be calculated using the formula:
\[ x_{cm} = \frac{m_1 x_1 - m_2 x_2}{m_1 - m_2} \]
Where:
- \(x_1 = 0\) (the larger disc's center)
- \(x_2 = R\) (the smaller disc's center)

Substituting the values:
\[ x_{cm} = \frac{(4\pi R^2 \sigma)(0) - (\pi R^2 \sigma)(R)}{4\pi R^2 \sigma - \pi R^2 \sigma} \]

### Step 4: Simplify the Expression
This simplifies to:
\[ x_{cm} = \frac{-\pi R^2 \sigma R}{(4\pi R^2 \sigma - \pi R^2 \sigma)} = \frac{-\pi R^3 \sigma}{3\pi R^2 \sigma} = \frac{-R}{3} \]

### Step 5: Interpret the Result
The negative sign indicates that the center of mass is located towards the center of the larger disc. Thus, the distance from the center of the larger disc is:
\[ \text{Distance from center} = \frac{R}{3} \]

### Step 6: Find the Value of \(\alpha\)
Since the problem states that the center of mass is \(\frac{\alpha}{R}\) from the center of the larger disc, we can equate:
\[ \frac{R}{3} = \frac{\alpha}{R} \]
Multiplying both sides by \(R\):
\[ \alpha = \frac{R^2}{3} \]

Thus, the value of \(\alpha\) is:
\[ \alpha = 1 \]

### Final Answer
The value of \(\alpha\) is \(1\).

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