A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 revolutions per minute (rpm). The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre___________.

Explanation

To solve the problem of a person moving from the rim to the center of a rotating circular platform, we can use the principle of conservation of angular momentum. Here’s the step-by-step solution:

### Step 1: Understand the Initial Conditions
- Mass of the person (m) = 80 kg
- Mass of the platform (M) = 200 kg
- Initial angular velocity of the platform (ω_initial) = 5 revolutions per minute (rpm)

### Step 2: Convert Angular Velocity to Radians per Second
To work with angular momentum, we often convert rpm to radians per second.
- 1 revolution = 2π radians
- Therefore, ω_initial in radians per second = (5 revolutions/minute) × (2π radians/revolution) × (1 minute/60 seconds) = (5 × 2π / 60) rad/s = (π / 6) rad/s.

### Step 3: Calculate the Initial Moment of Inertia
The moment of inertia (I_initial) of the system when the person is at the rim can be calculated as:
- Moment of inertia of the platform (I_platform) = (1/2) * M * R² (where R is the radius of the platform).
- Moment of inertia of the person at the rim = m * R².

Thus, the total initial moment of inertia:
\[ I_{initial} = I_{platform} + I_{person} = \frac{1}{2} M R^2 + m R^2 = \left(\frac{1}{2} \times 200 + 80\right) R^2 = (100 + 80) R^2 = 180 R^2. \]

### Step 4: Calculate Initial Angular Momentum
The initial angular momentum (L_initial) is given by:
\[ L_{initial} = I_{initial} \cdot ω_{initial} = 180 R^2 \cdot \frac{\pi}{6} = 30\pi R^2. \]

### Step 5: Consider the Final Moment of Inertia
When the person reaches the center of the platform, their moment of inertia becomes zero (since distance from the axis of rotation is zero). Therefore, the final moment of inertia (I_final) is just that of the platform:
\[ I_{final} = I_{platform} = \frac{1}{2} M R^2 = 100 R^2. \]

### Step 6: Apply Conservation of Angular Momentum
According to the conservation of angular momentum:
\[ L_{initial} = L_{final} \]
\[ 30\pi R^2 = I_{final} \cdot ω_{final} \]
\[ 30\pi R^2 = 100 R^2 \cdot ω_{final}. \]

### Step 7: Solve for Final Angular Velocity
Dividing both sides by R² (assuming R is not zero):
\[ 30\pi = 100 ω_{final} \]
\[ ω_{final} = \frac{30\pi}{100} = \frac{3\pi}{10} \text{ rad/s}. \]

### Step 8: Convert Final Angular Velocity Back to RPM
To convert back to rpm:
\[ ω_{final} = \frac{3\pi}{10} \text{ rad/s} \times \frac{60 \text{ seconds}}{2\pi \text{ radians}} = \frac{3 \times 60}{10 \times 2} = 9 \text{ rpm}. \]

### Final Answer
The rotational speed of the platform when the person reaches its center is **9 rpm**.