A

Step 1: Understand the system
We have a person of mass 80 kg standing on a circular platform of mass 200 kg. Initially, both the platform and the person are rotating together at 5 rpm.

Step 2: Apply the principle of conservation of angular momentum
The angular momentum of a system remains conserved if no external torque acts on it. The total initial angular momentum (L_initial) is:
$$L_{initial} = I_{initial} imes ext{angular velocity}_{initial}$$
Where $I_{initial}$ is the moment of inertia of the system.

Step 3: Calculate the moment of inertia
The moment of inertia of the person (treated as a point mass at a distance r, the radius of the platform) is:
$$I_{person} = m_{person} imes r^2 = 80 imes r^2$$
The moment of inertia of the platform is:
$$I_{platform} = m_{platform} imes R^2 = 200 imes R^2$$
Therefore, the total moment of inertia initially is:
$$I_{initial} = I_{person} + I_{platform} = 80 imes r^2 + 200 imes R^2$$
Since the person is standing on the rim initially, $r = R$:
$$I_{initial} = 80R^2 + 200R^2 = 280R^2$$

Step 4: Calculate initial angular momentum
Given that the initial angular velocity is 5 rpm, we convert it to radians per second:
$$ ext{angular velocity}_{initial} = 5 imes rac{2 ext{π}}{60} = rac{π}{6} ext{ rad/s}$$
Now, the initial angular momentum is:
$$L_{initial} = I_{initial} imes ext{angular velocity}_{initial} = 280R^2 imes rac{π}{6}$$

Step 5: Final scenario when the person is at the center
When the person moves to the center, the moment of inertia of the person becomes 0 (as $r = 0$):
$$I_{final} = I_{platform} = 200R^2$$
Let the final angular velocity be $ ext{angular velocity}_{final}$. The final angular momentum is:
$$L_{final} = I_{final} imes ext{angular velocity}_{final} = 200R^2 imes ext{angular velocity}_{final}$$

Step 6: Apply conservation of angular momentum
According to the conservation of angular momentum:
$$L_{initial} = L_{final}$$
Substituting, we get:
$$280R^2 imes rac{π}{6} = 200R^2 imes ext{angular velocity}_{final}$$
Now we can cancel $R^2$ (as long as $R eq 0$), giving us:
$$280 imes rac{π}{6} = 200 imes ext{angular velocity}_{final}$$

Step 7: Solve for final angular velocity
Rearranging gives:
$$ ext{angular velocity}_{final} = rac{280 imes rac{π}{6}}{200} = rac{280π}{1200} = rac{7π}{30} ext{ rad/s}$$

Step 8: Convert back to rpm
To convert the final angular velocity back to rpm:
$$ ext{angular velocity}_{final (rpm)} = rac{7π}{30} imes rac{60}{2π} = rac{7 imes 60}{30 imes 2} = rac{7 imes 1}{1} = 7 ext{ rpm}$$

Conclusion
Thus, the final rotational speed of the platform when the person reaches its center is 7 rpm.