Two discs of moments of inertia `I_1` and `I_2` about their respective axes , rotating with angular frequencies `omega_1` and `omega_2` respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be

Options

A. a. `I_(1)omega_1 + (I_(1)omega_2)/I_1 + I_2`
B. b. `I_(2)omega_1 + (I_(1)omega_2)/I_1 + I_2`
C. c. `I_(1)omega_1 - (I_(2)omega_2)/I_1 - I_2`
D. d. `I_(2)omega_1 - (I_(1)omega_2)/I_1 - I_2`

Explanation

To solve the problem of finding the angular frequency of the composite disc formed by two discs with given moments of inertia and angular frequencies, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Given Quantities**:
- Let the moments of inertia of the two discs be \( I_1 \) and \( I_2 \).
- Let the angular frequencies of the two discs be \( \omega_1 \) and \( \omega_2 \).

2. **Calculate the Initial Angular Momentum**:
- The total initial angular momentum \( L_{\text{initial}} \) of the system before the discs are brought into contact can be expressed as:
\[ L_{\text{initial}} = I_1 \omega_1 + I_2 \omega_2 \]

3. **Determine the Moment of Inertia of the Composite Disc**:
- When the two discs are brought into contact, they rotate together as a single composite disc. The total moment of inertia \( I_{\text{total}} \) of the composite disc is the sum of the individual moments of inertia:
\[ I_{\text{total}} = I_1 + I_2 \]

4. **Apply the Conservation of Angular Momentum**:
- According to the principle of conservation of angular momentum, the total initial angular momentum must equal the total final angular momentum after the discs are joined:
\[ L_{\text{initial}} = L_{\text{final}} \]
- The final angular momentum \( L_{\text{final}} \) can be expressed as:
\[ L_{\text{final}} = I_{\text{total}} \cdot \omega \]
- Therefore, we have:
\[ I_1 \omega_1 + I_2 \omega_2 = (I_1 + I_2) \cdot \omega \]

5. **Solve for the Final Angular Frequency \( \omega \)**:
- Rearranging the equation to solve for \( \omega \):
\[ \omega = \frac{I_1 \omega_1 + I_2 \omega_2}{I_1 + I_2} \]

6. **Final Result**:
- The angular frequency of the composite disc is given by:
\[ \omega = \frac{I_1 \omega_1 + I_2 \omega_2}{I_1 + I_2} \]