A disc of mass 1 kg and radius 0.1 m is rotating with angular velocity 20 rad/s. What is angular velocity (in rad/s) if a mass of 0.5 kg is put on the circumference of the disc ?

Options

A. 10
B. 20
C. 40
D. 30

Answer

To solve the problem, we will use the principle of conservation of angular momentum. The angular momentum before adding the mass must equal the angular momentum after adding the mass, assuming no external torques are acting on the system.

### Step-by-Step Solution:

1. **Calculate the Moment of Inertia of the Disc:**
The moment of inertia \( I \) of a disc about its central axis is given by the formula:
\[ I = \frac{1}{2} m r^2 \]
where \( m \) is the mass of the disc and \( r \) is its radius.

Given:
- Mass of the disc, \( m = 1 \, \text{kg} \)
- Radius of the disc, \( r = 0.1 \, \text{m} \)

Substituting the values:
\[ I_{\text{disc}} = \frac{1}{2} \times 1 \, \text{kg} \times (0.1 \, \text{m})^2 = \frac{1}{2} \times 1 \times 0.01 = 0.005 \, \text{kg m}^2 \]

2. **Calculate the Moment of Inertia of the Added Mass:**
When a mass \( m' = 0.5 \, \text{kg} \) is placed at the circumference of the disc, its moment of inertia about the same axis is given by:
\[ I' = m' r^2 \]
Substituting the values:
\[ I' = 0.5 \, \text{kg} \times (0.1 \, \text{m})^2 = 0.5 \times 0.01 = 0.005 \, \text{kg m}^2 \]

3. **Calculate the Total Moment of Inertia After Adding the Mass:**
The total moment of inertia \( I_{\text{total}} \) after adding the mass is:
\[ I_{\text{total}} = I_{\text{disc}} + I' = 0.005 \, \text{kg m}^2 + 0.005 \, \text{kg m}^2 = 0.01 \, \text{kg m}^2 \]

4. **Use Conservation of Angular Momentum:**
The initial angular momentum \( L_i \) is given by:
\[ L_i = I_{\text{disc}} \cdot \omega_1 \]
where \( \omega_1 = 20 \, \text{rad/s} \).

Substituting the values:
\[ L_i = 0.005 \, \text{kg m}^2 \cdot 20 \, \text{rad/s} = 0.1 \, \text{kg m}^2/\text{s} \]

The final angular momentum \( L_f \) is:
\[ L_f = I_{\text{total}} \cdot \omega_2 \]
where \( \omega_2 \) is the new angular velocity we want to find.

Setting initial and final angular momentum equal:
\[ L_i = L_f \implies 0.1 = 0.01 \cdot \omega_2 \]

5. **Solve for the New Angular Velocity \( \omega_2 \):**
Rearranging the equation gives:
\[ \omega_2 = \frac{0.1}{0.01} = 10 \, \text{rad/s} \]

### Final Answer:
The new angular velocity \( \omega_2 \) after adding the mass is \( 10 \, \text{rad/s} \).

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