A large disc has mass 2kg and radius 0.2 m and initial angular velocity 50 rad/s and small disc has mass 4kg and radius 0.1 m and initial angular velocity 200 rad/s both rotating about their common axis. Then the common final angular velocity after discs are in contact is,

Options

A. 100 rad/s
B. 125 rad/s
C. 200 rad/s
D. 150 rad/s

Explanation

To solve the problem of finding the common final angular velocity after the large and small discs come into contact, we can use the principle of conservation of angular momentum. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Given Data
- Mass of the large disc (m1) = 2 kg
- Radius of the large disc (r1) = 0.2 m
- Initial angular velocity of the large disc (ω1) = 50 rad/s
- Mass of the small disc (m2) = 4 kg
- Radius of the small disc (r2) = 0.1 m
- Initial angular velocity of the small disc (ω2) = 200 rad/s

### Step 2: Calculate the Moment of Inertia for Each Disc
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 large disc (I1):
\[ I_1 = \frac{1}{2} m_1 r_1^2 = \frac{1}{2} \times 2 \, \text{kg} \times (0.2 \, \text{m})^2 \]
\[ I_1 = \frac{1}{2} \times 2 \times 0.04 = 0.04 \, \text{kg m}^2 \]

For the small disc (I2):
\[ I_2 = \frac{1}{2} m_2 r_2^2 = \frac{1}{2} \times 4 \, \text{kg} \times (0.1 \, \text{m})^2 \]
\[ I_2 = \frac{1}{2} \times 4 \times 0.01 = 0.02 \, \text{kg m}^2 \]

### Step 3: Calculate the Initial Angular Momentum
The initial angular momentum (L_initial) of the system is the sum of the angular momentum of both discs:
\[ L_{\text{initial}} = I_1 \omega_1 + I_2 \omega_2 \]
\[ L_{\text{initial}} = 0.04 \, \text{kg m}^2 \times 50 \, \text{rad/s} + 0.02 \, \text{kg m}^2 \times 200 \, \text{rad/s} \]
\[ L_{\text{initial}} = 2.0 \, \text{kg m}^2/\text{s} + 4.0 \, \text{kg m}^2/\text{s} = 6.0 \, \text{kg m}^2/\text{s} \]

### Step 4: Apply Conservation of Angular Momentum
When the discs come into contact, they will rotate together with a common final angular velocity (ω_f). The final moment of inertia (I_f) of the system is the sum of the individual moments of inertia:
\[ I_f = I_1 + I_2 = 0.04 \, \text{kg m}^2 + 0.02 \, \text{kg m}^2 = 0.06 \, \text{kg m}^2 \]

Using the conservation of angular momentum:
\[ L_{\text{initial}} = L_{\text{final}} \]
\[ 6.0 \, \text{kg m}^2/\text{s} = I_f \omega_f \]
\[ 6.0 = 0.06 \omega_f \]

### Step 5: Solve for the Final Angular Velocity
Now, we can solve for ω_f:
\[ \omega_f = \frac{6.0}{0.06} = 100 \, \text{rad/s} \]

### Final Answer
The common final angular velocity after the discs are in contact is:
\[ \omega_f = 100 \, \text{rad/s} \]

---