A thin circular ring of mass m and radius R is rotating about its axis perpendicular to the plane of the ring with a constant angular velocity `omega`. Two point particleseach of mass M are attached gently to the opposite end of a diameter of the ring. The ring now rotates, with an angular velocity` (omega)/2`. Then the ratio`m/M` is

Options

A. 1
B. 2
C. `1/2`
D. `sqrt(2)`

Answer

To solve the problem, we will use the principle of conservation of angular momentum. Here are the steps to find the ratio \( \frac{m}{M} \):

### Step 1: Understand the Initial Conditions
Initially, we have a thin circular ring of mass \( m \) and radius \( R \) rotating with an angular velocity \( \omega \). The moment of inertia \( I \) of the ring about its axis is given by:
\[ I_{\text{ring}} = mR^2 \]

### Step 2: Calculate Initial Angular Momentum
The initial angular momentum \( L_i \) of the system can be calculated using the formula:
\[ L_i = I_{\text{ring}} \cdot \omega = mR^2 \cdot \omega \]

### Step 3: Understand the Final Conditions
After attaching two point masses \( M \) at opposite ends of a diameter of the ring, the system now rotates with an angular velocity of \( \frac{\omega}{2} \).

### Step 4: Calculate Final Moment of Inertia
The final moment of inertia \( I_f \) of the entire system (ring + two point masses) is:
\[ I_f = I_{\text{ring}} + 2 \cdot I_{\text{particle}} \]
Where \( I_{\text{particle}} = M R^2 \) (moment of inertia of each point mass about the axis). Thus:
\[ I_f = mR^2 + 2 \cdot (M R^2) = mR^2 + 2MR^2 \]

### Step 5: Calculate Final Angular Momentum
The final angular momentum \( L_f \) of the system is given by:
\[ L_f = I_f \cdot \omega_f = (mR^2 + 2MR^2) \cdot \frac{\omega}{2} \]

### Step 6: Set Initial and Final Angular Momentum Equal
By conservation of angular momentum, we have:
\[ L_i = L_f \]
Substituting the expressions we derived:
\[ mR^2 \cdot \omega = \left(mR^2 + 2MR^2\right) \cdot \frac{\omega}{2} \]

### Step 7: Cancel Common Factors
We can cancel \( R^2 \) and \( \omega \) from both sides (assuming \( R \neq 0 \) and \( \omega \neq 0 \)):
\[ m = \left(m + 2M\right) \cdot \frac{1}{2} \]

### Step 8: Rearrange the Equation
Multiplying both sides by 2 gives:
\[ 2m = m + 2M \]
Rearranging this, we get:
\[ 2m - m = 2M \]
Thus:
\[ m = 2M \]

### Step 9: Find the Ratio \( \frac{m}{M} \)
Now we can find the ratio:
\[ \frac{m}{M} = \frac{2M}{M} = 2 \]

### Final Answer
The ratio \( \frac{m}{M} \) is:
\[ \frac{m}{M} = 2 \]