A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now the platform is given an angular velocity `omega_(0)`. When the tortoise move along a chord of the platform with a constant velocity (with respect to the platform),

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of the tortoise on the rotating platform and how it affects the angular velocity of the system. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the System
The system consists of a circular platform rotating with an initial angular velocity \( \omega_0 \). A tortoise is sitting at the edge of the platform and moves along a chord with a constant velocity \( v \) relative to the platform.

### Step 2: Angular Momentum Conservation
Since there are no external torques acting on the system, the angular momentum of the system must be conserved. The total angular momentum \( L \) can be expressed as:
\[ L = I_{platform} \cdot \omega + I_{tortoise} \cdot \omega \]
where \( I_{platform} \) is the moment of inertia of the platform, \( I_{tortoise} \) is the moment of inertia of the tortoise, and \( \omega \) is the angular velocity of the system at any time.

### Step 3: Moment of Inertia
The moment of inertia of the tortoise changes as it moves along the chord. If the tortoise is at a distance \( r \) from the center of the platform, the moment of inertia of the tortoise can be expressed as:
\[ I_{tortoise} = m \cdot r^2 \]
where \( m \) is the mass of the tortoise.

### Step 4: Relationship Between Angular Velocity and Moment of Inertia
From the conservation of angular momentum, we have:
\[ I_{platform} \cdot \omega_0 + I_{tortoise} \cdot \omega_0 = I_{platform} \cdot \omega + I_{tortoise} \cdot \omega \]
This implies that as the tortoise moves and changes its distance \( r \), the angular velocity \( \omega \) must adjust to conserve angular momentum.

### Step 5: Analyzing the Motion
1. When the tortoise is at the edge of the platform, \( r \) is maximum, and thus \( I_{tortoise} \) is maximum. This results in a minimum angular velocity \( \omega \).
2. As the tortoise moves towards the center, \( r \) decreases, which decreases \( I_{tortoise} \) and increases \( \omega \).
3. When the tortoise reaches the center, \( I_{tortoise} \) is at its minimum, resulting in a maximum angular velocity \( \omega \).
4. As the tortoise moves back towards the edge, \( I_{tortoise} \) increases again, leading to a decrease in \( \omega \).

### Step 6: Conclusion
The relationship between angular velocity \( \omega \) and the position of the tortoise shows that \( \omega \) will have a minimum value when the tortoise is at the edge, a maximum value when it is at the center, and then decrease again as it moves back to the edge. This results in a graph of \( \omega \) versus the position of the tortoise that is a wave-like function, indicating the minimum and maximum values.

### Final Answer
The correct option representing the relationship between angular velocity \( \omega \) and the position of the tortoise is option C.

---