A round disc of moment of inertia `I_2` about its axis perpendicular to its plane and passing through its centre is placed over another disc of moment of inertia `I_1` rotating with an angular velocity `omega` about the same axis. The final angular velocity of the combination of discs is.

Options

A. `(I_2 omega)/(I_1 + I_2)`
B. `omega`
C. `(I_1 omega)/(I_1 + I_2)`
D. `((I_1 + I_2)omega)/(I_1)`

Explanation

To find the final angular velocity of the combination of the two discs, we can use the principle of conservation of angular momentum. The total angular momentum before the interaction must equal the total angular momentum after the interaction, assuming no external torques are acting on the system.

### Step-by-Step Solution:

1. **Identify the Initial Angular Momentum**:
The initial angular momentum \( L_i \) of the system is due to the rotating disc with moment of inertia \( I_1 \) and angular velocity \( \omega \):
\[ L_i = I_1 \cdot \omega \]

2. **Identify the Final Angular Momentum**:
After the second disc (with moment of inertia \( I_2 \)) is placed on top of the first disc, the total moment of inertia of the system becomes \( I_1 + I_2 \). Let the final angular velocity of the combined system be \( \omega_f \). The final angular momentum \( L_f \) is given by:
\[ L_f = (I_1 + I_2) \cdot \omega_f \]

3. **Apply Conservation of Angular Momentum**:
According to the conservation of angular momentum:
\[ L_i = L_f \]
Substituting the expressions for \( L_i \) and \( L_f \):
\[ I_1 \cdot \omega = (I_1 + I_2) \cdot \omega_f \]

4. **Solve for the Final Angular Velocity \( \omega_f \)**:
Rearranging the equation to solve for \( \omega_f \):
\[ \omega_f = \frac{I_1 \cdot \omega}{I_1 + I_2} \]

### Final Result:
The final angular velocity of the combination of discs is:
\[ \omega_f = \frac{I_1 \cdot \omega}{I_1 + I_2} \]