To find the common final angular velocity after the two discs are in contact, we can use the principle of conservation of angular momentum. Here’s a step-by-step solution:
### Step 1: Identify the given values
- Mass of the large disc, \( m_1 = 2 \, \text{kg} \)
- Radius of the large disc, \( r_1 = 0.2 \, \text{m} \)
- Initial angular velocity of the large disc, \( \omega_1 = 50 \, \text{rad/s} \)
- Mass of the small disc, \( m_2 = 4 \, \text{kg} \)
- Radius of the small disc, \( r_2 = 0.1 \, \text{m} \)
- Initial angular velocity of the small disc, \( \omega_2 = 200 \, \text{rad/s} \)
### Step 2: Calculate the moment of inertia for both discs
The moment of inertia \( I \) for a disc rotating about its central axis is given by the formula:
\[
I = \frac{1}{2} m r^2
\]
**For the small disc:**
\[
I_2 = \frac{1}{2} m_2 r_2^2 = \frac{1}{2} \times 4 \times (0.1)^2 = \frac{1}{2} \times 4 \times 0.01 = 0.02 \, \text{kg m}^2
\]
**For the large disc:**
\[
I_1 = \frac{1}{2} m_1 r_1^2 = \frac{1}{2} \times 2 \times (0.2)^2 = \frac{1}{2} \times 2 \times 0.04 = 0.04 \, \text{kg m}^2
\]
### Step 3: Calculate the initial angular momentum of both discs
The initial angular momentum \( L \) is given by:
\[
L = I \omega
\]
**For the small disc:**
\[
L_2 = I_2 \omega_2 = 0.02 \times 200 = 4 \, \text{kg m}^2/\text{s}
\]
**For the large disc:**
\[
L_1 = I_1 \omega_1 = 0.04 \times 50 = 2 \, \text{kg m}^2/\text{s}
\]
### Step 4: Calculate the total initial angular momentum
\[
L_{\text{initial}} = L_1 + L_2 = 2 + 4 = 6 \, \text{kg m}^2/\text{s}
\]
### Step 5: Calculate the total moment of inertia after the discs are in contact
\[
I_{\text{total}} = I_1 + I_2 = 0.04 + 0.02 = 0.06 \, \text{kg m}^2
\]
### Step 6: Apply conservation of angular momentum
According to the conservation of angular momentum:
\[
L_{\text{initial}} = L_{\text{final}}
\]
\[
L_{\text{final}} = I_{\text{total}} \omega_f
\]
Thus,
\[
6 = 0.06 \omega_f
\]
### Step 7: Solve for the final angular velocity \( \omega_f \)
\[
\omega_f = \frac{6}{0.06} = 100 \, \text{rad/s}
\]
### Final Answer
The common final angular velocity after the discs are in contact is \( \omega_f = 100 \, \text{rad/s} \).
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