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. `omega`
B. `(I_1omega)/(I_1+I_2)`
C. `((I_1+I_2)omega)/I_1`
D. `(I_2omega)/(I_1+I_2)`

Explanation

To solve the problem, we need to apply the principle of conservation of angular momentum. Here’s a step-by-step solution:

### Step 1: Understand the System
We have two discs:
- Disc 1 with moment of inertia \( I_1 \) rotating with an angular velocity \( \omega \).
- Disc 2 with moment of inertia \( I_2 \) which is initially at rest (angular velocity = 0).

### Step 2: Calculate Initial Angular Momentum
The initial angular momentum \( L_i \) of the system can be calculated as follows:
\[ L_i = I_1 \cdot \omega + I_2 \cdot 0 = I_1 \cdot \omega \]
Thus, the initial angular momentum is simply \( I_1 \cdot \omega \).

### Step 3: Calculate Final Angular Momentum
After the two discs are combined, they will rotate together with a new angular velocity \( \omega' \). The final angular momentum \( L_f \) of the system is given by:
\[ L_f = (I_1 + I_2) \cdot \omega' \]

### Step 4: Apply Conservation of Angular Momentum
According to the conservation of angular momentum, the initial angular momentum must equal the final angular momentum:
\[ L_i = L_f \]
Substituting the expressions we derived:
\[ I_1 \cdot \omega = (I_1 + I_2) \cdot \omega' \]

### Step 5: Solve for Final Angular Velocity
To find the final angular velocity \( \omega' \), we rearrange the equation:
\[ \omega' = \frac{I_1 \cdot \omega}{I_1 + I_2} \]

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