A gramophone record of mass M and radius R is rotating at an angular velocity `omega` A coin of mass is gently placed on the record at a distance `r = (R )/(2)` from its centre. The new angular velocity of the system is

Options

A. `(2omegaM)/((2M+m))`
B. `(2omegaM)/((M+2m))`
C. `omega`
D. `(omegaM)/(m)`

Explanation

To solve the problem of finding the new angular velocity of the gramophone record after a coin is placed on it, we can use the principle of conservation of angular momentum. Here’s a step-by-step solution:

### Step 1: Understand the Initial Conditions
The gramophone record has:
- Mass = M
- Radius = R
- Initial angular velocity = ω

The coin has:
- Mass = m
- It is placed at a distance r = R/2 from the center of the record.

### Step 2: Calculate the Moment of Inertia of the System
The moment of inertia (I) of the gramophone record (considered as a solid disk) about its center is given by:
\[ I_{\text{record}} = \frac{1}{2} M R^2 \]

The moment of inertia of the coin when it is placed at a distance r = R/2 from the center is:
\[ I_{\text{coin}} = m \left(\frac{R}{2}\right)^2 = m \frac{R^2}{4} \]

### Step 3: Calculate the Total Initial Moment of Inertia
The total initial moment of inertia of the system (record + coin) before the coin is placed is:
\[ I_{\text{initial}} = I_{\text{record}} = \frac{1}{2} M R^2 \]

### Step 4: Calculate the Total Final Moment of Inertia
After placing the coin, the total moment of inertia becomes:
\[ I_{\text{final}} = I_{\text{record}} + I_{\text{coin}} = \frac{1}{2} M R^2 + m \frac{R^2}{4} \]

### Step 5: Apply Conservation of Angular Momentum
Since there is no external torque acting on the system, the angular momentum before and after placing the coin must be conserved:
\[ I_{\text{initial}} \cdot \omega = I_{\text{final}} \cdot \omega' \]
Where ω' is the new angular velocity.

Substituting the values we have:
\[ \left(\frac{1}{2} M R^2\right) \cdot \omega = \left(\frac{1}{2} M R^2 + m \frac{R^2}{4}\right) \cdot \omega' \]

### Step 6: Solve for the New Angular Velocity (ω')
Rearranging the equation to solve for ω':
\[ \omega' = \frac{\left(\frac{1}{2} M R^2\right) \cdot \omega}{\left(\frac{1}{2} M R^2 + m \frac{R^2}{4}\right)} \]

### Step 7: Simplify the Expression
Factoring out \( R^2 \) from the denominator:
\[ \omega' = \frac{\frac{1}{2} M \omega}{\frac{1}{2} M + \frac{m}{4}} \]

### Final Expression
Thus, the new angular velocity of the system after placing the coin is:
\[ \omega' = \frac{2M\omega}{2M + m} \]