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 solution:

### Step 1: Understand the System
We have two discs with moments of inertia \( I_1 \) and \( I_2 \) respectively. They are rotating about the same axis but in opposite directions with angular speeds \( \omega_1 \) and \( \omega_2 \).

### Step 2: Write the Angular Momentum of Each Disc
The angular momentum \( L \) of a rotating object is given by the product of its moment of inertia and its angular velocity. Therefore, the angular momentum of the first disc is:
\[ L_1 = I_1 \omega_1 \]
And for the second disc, since it is rotating in the opposite direction, its angular momentum is:
\[ L_2 = -I_2 \omega_2 \]

### Step 3: Apply Conservation of Angular Momentum
According to the conservation of angular momentum, the total angular momentum before they are combined must equal the total angular momentum after they are combined. Thus, we write:
\[ L_{\text{total}} = L_1 + L_2 = I_1 \omega_1 - I_2 \omega_2 \]

### Step 4: Write the Total Moment of Inertia of the Combined System
When the two discs are combined, their moments of inertia add up:
\[ I_{\text{total}} = I_1 + I_2 \]

### Step 5: Set Up the Equation for Final Angular Velocity
Let \( \omega_f \) be the final angular velocity of the combined system. The angular momentum of the combined system can also be expressed as:
\[ L_{\text{total}} = I_{\text{total}} \omega_f = (I_1 + I_2) \omega_f \]

### Step 6: Equate the Angular Momenta
Setting the total angular momentum before and after the combination equal gives us:
\[ I_1 \omega_1 - I_2 \omega_2 = (I_1 + I_2) \omega_f \]

### Step 7: Solve for Final Angular Velocity
Rearranging the equation to solve for \( \omega_f \):
\[ \omega_f = \frac{I_1 \omega_1 - I_2 \omega_2}{I_1 + I_2} \]

### Step 8: Find the Magnitude of Angular Velocity
Since we are interested in the magnitude of angular velocity, we take the absolute value:
\[ |\omega_f| = \left| \frac{I_1 \omega_1 - I_2 \omega_2}{I_1 + I_2} \right| \]

### Final Answer
Thus, the magnitude of the angular velocity of the system is:
\[ |\omega_f| = \frac{|I_1 \omega_1 - I_2 \omega_2|}{I_1 + I_2} \]

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