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 step-by-step calculations:

### Step 1: Understand the Initial Condition
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 is given by:
\[ I_{\text{ring}} = mR^2 \]

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

### Step 3: Understand the Final Condition
When two point masses \( M \) are attached to the ring at opposite ends of a diameter, the new angular velocity becomes \( \frac{\omega}{2} \). The new moment of inertia \( I_f \) of the system (ring + two masses) is:
\[ I_f = I_{\text{ring}} + I_{\text{masses}} = mR^2 + 2M \cdot R^2 \]
Here, \( 2M \cdot R^2 \) accounts for the two point masses located at a distance \( R \) from the axis of rotation.

### Step 4: Calculate Final Angular Momentum
The final angular momentum \( L_f \) of the system can be calculated as:
\[ L_f = I_f \cdot \omega' = (mR^2 + 2MR^2) \cdot \frac{\omega}{2} \]

### Step 5: Set Initial and Final Angular Momentum Equal
According to the conservation of angular momentum:
\[ L_i = L_f \]
Substituting the expressions we derived:
\[ mR^2 \cdot \omega = \left(mR^2 + 2MR^2\right) \cdot \frac{\omega}{2} \]

### Step 6: Simplify the Equation
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 7: Multiply Both Sides by 2
To eliminate the fraction, multiply both sides by 2:
\[ 2m = m + 2M \]

### Step 8: Rearrange the Equation
Rearranging gives:
\[ 2m - m = 2M \implies m = 2M \]

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

### Conclusion
The ratio \( \frac{m}{M} \) is \( 2 \).

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