A rotating table completes one rotation is 10 sec. and its moment of ineratia is 100 kg-`m^2`. A person of 50 kg. mass stands at the centre of the rotating table. If the person moves 2m. from the centre, the angular velocity of the rotating table in rad/sec. will be:

Options

A. `(2pi)/(3 0)`
B. `(20 pi)/(30)`
C. `(2pi)/3`
D. `2 pi `

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Determine the initial angular velocity (ω₁)
The table completes one rotation in 10 seconds. The angular velocity (ω) can be calculated using the formula:

\[ \omega = \frac{2\pi}{T} \]

where \(T\) is the time period for one complete rotation.

Substituting \(T = 10\) seconds:

\[ \omega_1 = \frac{2\pi}{10} = \frac{\pi}{5} \text{ rad/sec} \]

### Step 2: Calculate the initial moment of inertia (I₁)
The initial moment of inertia (I₁) of the rotating table is given as:

\[ I_1 = 100 \text{ kg m}^2 \]

### Step 3: Calculate the new moment of inertia (I₂) when the person moves
When the person of mass \(m = 50\) kg moves 2 meters from the center, the new moment of inertia (I₂) can be calculated using the formula:

\[ I_2 = I_1 + m \cdot r^2 \]

where \(r\) is the distance moved (2 m).

Substituting the values:

\[ I_2 = 100 + 50 \cdot (2^2) = 100 + 50 \cdot 4 = 100 + 200 = 300 \text{ kg m}^2 \]

### Step 4: Apply the conservation of angular momentum
According to the conservation of angular momentum, we have:

\[ I_1 \omega_1 = I_2 \omega_2 \]

We need to find the new angular velocity (ω₂). Rearranging the equation gives:

\[ \omega_2 = \frac{I_1 \omega_1}{I_2} \]

### Step 5: Substitute the known values to find ω₂
Substituting \(I_1 = 100\), \(\omega_1 = \frac{\pi}{5}\), and \(I_2 = 300\):

\[ \omega_2 = \frac{100 \cdot \frac{\pi}{5}}{300} \]

Calculating this gives:

\[ \omega_2 = \frac{100\pi}{1500} = \frac{2\pi}{30} \text{ rad/sec} \]

### Final Result
Thus, the angular velocity of the rotating table after the person moves 2 meters from the center is:

\[ \omega_2 = \frac{2\pi}{30} \text{ rad/sec} \]

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