Initial angular velocity of a circular disc of mass `M` is `omega_(1)`. Then two small spheres of mass `m` are attached gently to two diametrically opposite points on the edge of the disc. What is the final angular velocity of the disc -

Options

A. `((M+m)/M)omega_(1)`
B. `((M+m)/m)omega_(1)`
C. `(M/(M+4m))omega_(1)`
D. `(M/(M+2m))omega_(1)`

Explanation

To solve the problem of finding the final angular velocity of a circular disc after two small spheres are attached to it, we will use the principle of conservation of angular momentum.

### Step-by-Step Solution:

1. **Identify Initial Conditions:**
- The initial angular velocity of the disc is \( \omega_1 \).
- The mass of the disc is \( M \).
- The radius of the disc is \( R \).
- The mass of each small sphere is \( m \).

2. **Calculate Initial Moment of Inertia:**
- The moment of inertia \( I_1 \) of the disc about its center is given by:
\[ I_1 = \frac{1}{2} M R^2 \]

3. **Determine the Moment of Inertia of the Spheres:**
- When the two small spheres are attached to the disc, they are located at a distance \( R \) from the center. The moment of inertia \( I_2 \) for one sphere is:
\[ I_2 = m R^2 \]
- Since there are two spheres, the total moment of inertia contributed by the spheres is:
\[ I_{spheres} = 2 \cdot m R^2 = 2m R^2 \]

4. **Calculate Total Moment of Inertia After Spheres are Added:**
- The total moment of inertia \( I_{total} \) after adding the spheres is:
\[ I_{total} = I_1 + I_{spheres} = \frac{1}{2} M R^2 + 2m R^2 \]

5. **Apply Conservation of Angular Momentum:**
- According to the conservation of angular momentum:
\[ I_1 \omega_1 = I_{total} \omega_{final} \]
- Substituting the values we have:
\[ \frac{1}{2} M R^2 \omega_1 = \left(\frac{1}{2} M R^2 + 2m R^2\right) \omega_{final} \]

6. **Solve for Final Angular Velocity \( \omega_{final} \):**
- Rearranging the equation gives:
\[ \omega_{final} = \frac{\frac{1}{2} M R^2 \omega_1}{\frac{1}{2} M R^2 + 2m R^2} \]
- Simplifying this expression:
\[ \omega_{final} = \frac{M \omega_1}{M + 4m} \]

### Final Answer:
Thus, the final angular velocity of the disc after the two small spheres are attached is:
\[ \omega_{final} = \frac{M}{M + 4m} \omega_1 \]