A ring is set to rotate with angular speed `omega_(0)` in gravity-free space. Ring is made from flexible material and due to centrifugal action its radius starts increasing slowly. After some time radius of ring becomes twice of its initial value and its angular velocity is found to be `omega` . Calculate `omega_(0)// omega`.

Explanation

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

### Step 1: Understand the initial and final conditions
- The ring is initially rotating with an angular speed \( \omega_0 \) and has an initial radius \( r \).
- After some time, the radius of the ring doubles, so the final radius is \( 2r \).
- The final angular speed is \( \omega \).

### Step 2: Write down the expression for angular momentum
The angular momentum \( L \) of a rotating object is given by the formula:
\[ L = I \cdot \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular speed.

### Step 3: Calculate the initial angular momentum
The moment of inertia \( I \) of a ring about its center is given by:
\[ I = M r^2 \]
Thus, the initial angular momentum \( L_0 \) is:
\[ L_0 = I_0 \cdot \omega_0 = M r^2 \cdot \omega_0 \]

### Step 4: Calculate the final angular momentum
After the radius doubles, the new moment of inertia \( I_f \) becomes:
\[ I_f = M (2r)^2 = M \cdot 4r^2 \]
Thus, the final angular momentum \( L_f \) is:
\[ L_f = I_f \cdot \omega = 4M r^2 \cdot \omega \]

### Step 5: Apply the conservation of angular momentum
Since there are no external torques acting on the system, the angular momentum is conserved:
\[ L_0 = L_f \]
Substituting the expressions we derived:
\[ M r^2 \cdot \omega_0 = 4M r^2 \cdot \omega \]

### Step 6: Simplify the equation
We can cancel \( M \) and \( r^2 \) from both sides (assuming \( M \neq 0 \) and \( r \neq 0 \)):
\[ \omega_0 = 4 \omega \]

### Step 7: Find the ratio \( \frac{\omega_0}{\omega} \)
Rearranging the equation gives us:
\[ \frac{\omega_0}{\omega} = 4 \]

### Final Answer
Thus, the ratio \( \frac{\omega_0}{\omega} \) is \( 4 \).

---