A disc of moment of inertia `2 kg -m^(23)` revolving with `8 rad//s` is placed on anothere disc of moment of inertia `4 kg-m^(2)` revolving `4 rad//s`.What is the angular frequency of composite disc?

Options

A. `4rad//s`
B. `3/16rad//s`
C. `16/3rad//s`
D. `16/5rad//s`

Answer

To solve the problem, we will use the principle of conservation of angular momentum. The total angular momentum before the two discs interact must equal the total angular momentum after they have combined.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Moment of inertia of the first disc, \( I_1 = 2 \, \text{kg m}^2 \)
- Angular velocity of the first disc, \( \omega_1 = 8 \, \text{rad/s} \)
- Moment of inertia of the second disc, \( I_2 = 4 \, \text{kg m}^2 \)
- Angular velocity of the second disc, \( \omega_2 = 4 \, \text{rad/s} \)

2. **Calculate the Initial Angular Momentum:**
- The initial angular momentum of the first disc:
\[ L_1 = I_1 \cdot \omega_1 = 2 \cdot 8 = 16 \, \text{kg m}^2/\text{s} \]
- The initial angular momentum of the second disc:
\[ L_2 = I_2 \cdot \omega_2 = 4 \cdot 4 = 16 \, \text{kg m}^2/\text{s} \]
- Total initial angular momentum:
\[ L_{\text{initial}} = L_1 + L_2 = 16 + 16 = 32 \, \text{kg m}^2/\text{s} \]

3. **Calculate the Total Moment of Inertia After Combining:**
- The total moment of inertia of the composite disc:
\[ I_{\text{total}} = I_1 + I_2 = 2 + 4 = 6 \, \text{kg m}^2 \]

4. **Apply Conservation of Angular Momentum:**
- According to the conservation of angular momentum:
\[ L_{\text{initial}} = L_{\text{final}} \]
- The final angular momentum of the composite disc is given by:
\[ L_{\text{final}} = I_{\text{total}} \cdot \omega \]
- Setting the initial and final angular momentum equal:
\[ 32 = 6 \cdot \omega \]

5. **Solve for the Final Angular Velocity \( \omega \):**
- Rearranging the equation to find \( \omega \):
\[ \omega = \frac{32}{6} = \frac{16}{3} \, \text{rad/s} \]

### Final Answer:
The angular frequency of the composite disc is \( \frac{16}{3} \, \text{rad/s} \).

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