A circular platform is mounted on a vertical frictionless axle. Its radius is `r = 2 m` and its moment of inertia`I = 200 kg m^2`. It is initially at rest. A `70 kg` man stands on the edge of the platform and begins to walk along the edge at speed `v_0 = 1 m s^-1` relative to the ground. The angular velocity of the platform is: a) 1.2 rad/s b) 0.4 rad/s c) 0.7 rad/s d) 2 rad/s

Options

A. `1.2 rad s^(-1)`
B. `0.4 rad s^(-1)`
C. `0.7 rad s^(-1)`
D. `2 rad s^(-1)`

Explanation

To solve the problem, we will use the principle of conservation of angular momentum.

### Step-by-Step Solution:

1. **Understand the System**:
- We have a circular platform with a radius \( r = 2 \, \text{m} \) and a moment of inertia \( I = 200 \, \text{kg m}^2 \).
- A man with mass \( m = 70 \, \text{kg} \) walks at a speed \( v_0 = 1 \, \text{m/s} \) along the edge of the platform.

2. **Initial Conditions**:
- The platform is initially at rest, so its initial angular momentum is \( L_i = 0 \).

3. **Final Angular Momentum**:
- When the man walks on the edge of the platform, he has angular momentum due to his motion. The angular momentum of the man can be calculated using the formula:
\[ L_{\text{man}} = m \cdot v_0 \cdot r \]
- Substituting the known values:
\[ L_{\text{man}} = 70 \, \text{kg} \cdot 1 \, \text{m/s} \cdot 2 \, \text{m} = 140 \, \text{kg m}^2/\text{s} \]

4. **Conservation of Angular Momentum**:
- According to the conservation of angular momentum, the total angular momentum before the man starts walking must equal the total angular momentum after he starts walking:
\[ L_i + L_{\text{platform}} = L_f \]
- Since the initial angular momentum \( L_i = 0 \) and the platform has no initial angular momentum, we have:
\[ 0 = L_{\text{man}} + I \cdot \omega \]
- Rearranging gives:
\[ I \cdot \omega = -L_{\text{man}} \]

5. **Calculate Angular Velocity**:
- Substituting the values:
\[ 200 \, \text{kg m}^2 \cdot \omega = -140 \, \text{kg m}^2/\text{s} \]
- Solving for \( \omega \):
\[ \omega = \frac{-140}{200} = -0.7 \, \text{rad/s} \]
- The negative sign indicates that the platform rotates in the opposite direction to the man's walking direction.

6. **Final Answer**:
- The magnitude of the angular velocity of the platform is \( 0.7 \, \text{rad/s} \). Therefore, the correct option is **c) 0.7 rad/s**.