A disc of mass 2 kg and radius 0.2 m is rotating with angular veocity `"30 rad s"^(-1)`. What is angular velocity, if a mass of 0.25 kg is put on periphery of the disc?

Options

A. `"24 rad s"^(-1)`
B. `"36 rad s"^(-1)`
C. `"15 rad s"^(-1)`
D. `"26 rad s"^(-1)`

Explanation

To solve the problem, we need to apply the principle of conservation of angular momentum. The initial angular momentum of the system must equal the final angular momentum after the mass is added to the disc.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mass of the disc (M) = 2 kg
- Radius of the disc (r) = 0.2 m
- Initial angular velocity (ω_initial) = 30 rad/s
- Mass added at the periphery (m) = 0.25 kg

2. **Calculate the Initial Moment of Inertia (I_initial) of the Disc:**
The moment of inertia of a disc about its center is given by:
\[ I_{initial} = \frac{1}{2} M r^2 \]
Substituting the values:
\[ I_{initial} = \frac{1}{2} \times 2 \, \text{kg} \times (0.2 \, \text{m})^2 = \frac{1}{2} \times 2 \times 0.04 = 0.04 \, \text{kg m}^2 \]

3. **Calculate the Moment of Inertia (I_final) After Adding the Mass:**
The moment of inertia of the point mass at the periphery of the disc is given by:
\[ I_{point} = m r^2 \]
Therefore, the total final moment of inertia is:
\[ I_{final} = I_{initial} + I_{point} = I_{initial} + m r^2 \]
Substituting the values:
\[ I_{point} = 0.25 \, \text{kg} \times (0.2 \, \text{m})^2 = 0.25 \times 0.04 = 0.01 \, \text{kg m}^2 \]
Now, adding this to the initial moment of inertia:
\[ I_{final} = 0.04 \, \text{kg m}^2 + 0.01 \, \text{kg m}^2 = 0.05 \, \text{kg m}^2 \]

4. **Apply Conservation of Angular Momentum:**
According to the conservation of angular momentum:
\[ L_{initial} = L_{final} \]
This can be expressed as:
\[ I_{initial} \cdot \omega_{initial} = I_{final} \cdot \omega_{final} \]
Substituting the known values:
\[ 0.04 \, \text{kg m}^2 \cdot 30 \, \text{rad/s} = 0.05 \, \text{kg m}^2 \cdot \omega_{final} \]

5. **Solve for the Final Angular Velocity (ω_final):**
Rearranging the equation to solve for ω_final:
\[ \omega_{final} = \frac{0.04 \cdot 30}{0.05} \]
Calculating this gives:
\[ \omega_{final} = \frac{1.2}{0.05} = 24 \, \text{rad/s} \]

### Final Answer:
The final angular velocity after adding the mass to the disc is **24 rad/s**.