If two disc of moment of inertia `l_(1)` and `l_(2)` rotating about collinear axis passing through their centres of mass and perpendicular to their plane with angular speeds `omega_(1)` and `omega_(2)` respectively in opposite directions are made to rotate combinedly along same axis, then the magnitude of angular velocity of the system is

Options

A. `(l_(1)omega_(1)+l_(2)omega_(2))/(l_(1)+l_(2))`
B. `(l_(1)omega_(1)-l_(2)omega_(2))/(l_(1)+l_(2))`
C. `(l_(1)omega_(1)+l_(2)omega_(2))/(omega_(1)+omega_(2))`
D. `(l_(1)omega_(1)-l_(2)omega_(2))/(omega_(1)-omega_(2))`

Answer

To solve the problem, we will use the principle of conservation of angular momentum. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the system
We have two discs with moments of inertia \( I_1 \) and \( I_2 \) rotating about the same axis. The first disc rotates with angular velocity \( \omega_1 \) and the second disc rotates with angular velocity \( \omega_2 \) in the opposite direction.

### Step 2: Define angular momentum for each disc
The angular momentum \( L \) of a rotating object is given by the formula:
\[ L = I \cdot \omega \]
For the first disc, the angular momentum \( L_1 \) is:
\[ L_1 = I_1 \cdot \omega_1 \]
For the second disc, since it is rotating in the opposite direction, we take its angular velocity as negative:
\[ L_2 = I_2 \cdot (-\omega_2) = -I_2 \cdot \omega_2 \]

### Step 3: Calculate the total initial angular momentum
The total initial angular momentum \( L_{\text{initial}} \) of the system is the sum of the angular momenta of both discs:
\[ L_{\text{initial}} = L_1 + L_2 = I_1 \cdot \omega_1 - I_2 \cdot \omega_2 \]

### Step 4: Combine the discs
When the two discs are combined, their moments of inertia add up:
\[ I_{\text{final}} = I_1 + I_2 \]
Let the final angular velocity of the combined system be \( \omega_f \).

### Step 5: Apply conservation of angular momentum
According to the conservation of angular momentum, the total initial angular momentum must equal the total final angular momentum:
\[ L_{\text{initial}} = L_{\text{final}} \]
Thus,
\[ I_1 \cdot \omega_1 - I_2 \cdot \omega_2 = (I_1 + I_2) \cdot \omega_f \]

### Step 6: Solve for the final angular velocity \( \omega_f \)
Rearranging the equation to solve for \( \omega_f \):
\[ \omega_f = \frac{I_1 \cdot \omega_1 - I_2 \cdot \omega_2}{I_1 + I_2} \]

### Conclusion
The magnitude of the angular velocity of the combined system is given by:
\[ \omega_f = \frac{I_1 \cdot \omega_1 - I_2 \cdot \omega_2}{I_1 + I_2} \]