A circular platform is mounted on a frictionless vertical axle. Its radius `R = 2m` and its moment of inertia about the axle is `200 kg m^(2)`. It is initially at rest. A `50 kg` man stands on the edge at the platform and begins to walk along the edge at the speed of `1ms^(-1)` relative to the ground. Time taken by the man to complete one revolution is :

Options

A. `2pi s `
B. `(pi)/(2)s`
C. `pi s `
D. `(3pi)/(2)s`

Answer

To solve the problem step by step, we will analyze the situation involving the circular platform and the man walking on it.

### Step 1: Understand the System
We have a circular platform with:
- Radius \( R = 2 \, \text{m} \)
- Moment of inertia \( I = 200 \, \text{kg m}^2 \)
- A man with mass \( m = 50 \, \text{kg} \) walking at a speed of \( v_m = 1 \, \text{m/s} \) relative to the ground.

### Step 2: Conservation of Angular Momentum
Since there is no external torque acting on the system, the total angular momentum is conserved.

The initial angular momentum of the system (when the man is at rest) is:
\[ L_{\text{initial}} = 0 \]

When the man starts walking, the angular momentum of the man and the platform must balance:
\[ L_{\text{final}} = I \omega + m v_m R \]
where \( \omega \) is the angular velocity of the platform.

### Step 3: Relate Angular Momentum of the Man and the Platform
The angular momentum of the man can be expressed as:
\[ L_{\text{man}} = m v_m R \]
Substituting the values:
\[ L_{\text{man}} = 50 \, \text{kg} \times 1 \, \text{m/s} \times 2 \, \text{m} = 100 \, \text{kg m}^2/\text{s} \]

The angular momentum of the platform is:
\[ L_{\text{platform}} = I \omega = 200 \, \text{kg m}^2 \cdot \omega \]

### Step 4: Set Up the Equation
Since the total angular momentum is conserved:
\[ 0 = 200 \omega + 100 \]
Solving for \( \omega \):
\[ 200 \omega = -100 \implies \omega = -0.5 \, \text{rad/s} \]

The negative sign indicates that the platform rotates in the opposite direction to the man's walking direction.

### Step 5: Calculate the Linear Velocity of the Platform
The linear velocity \( v_p \) of the platform can be calculated using:
\[ v_p = R \omega = 2 \, \text{m} \times -0.5 \, \text{rad/s} = -1 \, \text{m/s} \]

### Step 6: Find the Velocity of the Man Relative to the Platform
The velocity of the man relative to the platform is given by:
\[ v_{mp} = v_m - v_p = 1 \, \text{m/s} - (-1 \, \text{m/s}) = 1 + 1 = 2 \, \text{m/s} \]

### Step 7: Calculate the Time for One Revolution
The distance the man needs to cover to complete one revolution is the circumference of the circle:
\[ \text{Distance} = 2 \pi R = 2 \pi \times 2 = 4 \pi \, \text{m} \]

Now, using the relative velocity:
\[ \text{Time} = \frac{\text{Distance}}{\text{Velocity}} = \frac{4 \pi \, \text{m}}{2 \, \text{m/s}} = 2 \pi \, \text{s} \]

### Final Answer
The time taken by the man to complete one revolution is \( 2 \pi \, \text{s} \).

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