A wheel of moment of inertia `2 kgm^(2)` is rotating about an axis passing through centre and perpendicular to its plane at a speed `60 rad//s`. Due to friction, it comes to rest in 5 minutes. The angular momentum of the wheel three minutes before it stops rotating is

Options

A. `"24 kg m"^(2)//s`
B. `"48 kg m"^(2)//s`
C. `"72 kg m"^(2)//s`
D. `"96 kg m"^(2)//s`

Answer

To solve the problem step by step, we need to find the angular momentum of the wheel three minutes before it comes to rest.

### Step 1: Understand the given data
- Moment of inertia (I) = 2 kg·m²
- Initial angular speed (ω_initial) = 60 rad/s
- Time taken to come to rest = 5 minutes = 5 × 60 = 300 seconds
- Time before it stops when we need to find the angular momentum = 3 minutes = 3 × 60 = 180 seconds

### Step 2: Calculate the angular acceleration (α)
The angular acceleration can be calculated using the formula:
\[ \alpha = \frac{\omega_f - \omega_i}{t} \]
where:
- \(\omega_f\) = final angular velocity = 0 rad/s (since it comes to rest)
- \(\omega_i\) = initial angular velocity = 60 rad/s
- \(t\) = time taken to stop = 300 seconds

Substituting the values:
\[ \alpha = \frac{0 - 60}{300} = \frac{-60}{300} = -0.2 \text{ rad/s}^2 \]

### Step 3: Find the angular velocity at 2 minutes (or 120 seconds) before it stops
Since we need to find the angular velocity 3 minutes before it stops, we need to find it at \(t = 300 - 180 = 120\) seconds.

Using the formula:
\[ \omega = \omega_i + \alpha t \]
Substituting the values:
\[ \omega = 60 + (-0.2)(120) \]
Calculating:
\[ \omega = 60 - 24 = 36 \text{ rad/s} \]

### Step 4: Calculate the angular momentum (L) at that time
The angular momentum can be calculated using the formula:
\[ L = I \cdot \omega \]
Substituting the values:
\[ L = 2 \cdot 36 = 72 \text{ kg·m}^2/\text{s} \]

### Final Answer
The angular momentum of the wheel three minutes before it stops rotating is **72 kg·m²/s**.

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