A sphere of moment of inertia `10kgm^(2)` is rotating at a speed of 100 rad/s. Calculate the torque required to stop it in 10 minutes. Also calculate the angular momentum of the wheel two minutes before it stops rotating.

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the angular deceleration (α)
We know that the sphere is to stop in 10 minutes. First, we need to convert 10 minutes into seconds:
\[ 10 \text{ minutes} = 10 \times 60 = 600 \text{ seconds} \]

The initial angular velocity (ω₀) is given as 100 rad/s, and the final angular velocity (ω) when it stops is 0 rad/s. We can use the formula for angular acceleration (α):
\[ \alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega - \omega_0}{t} \]
Substituting the values:
\[ \alpha = \frac{0 - 100}{600} = \frac{-100}{600} = -\frac{1}{6} \text{ rad/s}^2 \]

### Step 2: Calculate the torque (τ)
Torque is related to angular acceleration and moment of inertia by the formula:
\[ \tau = I \cdot \alpha \]
Where:
- \( I = 10 \text{ kg m}^2 \) (moment of inertia)
- \( \alpha = -\frac{1}{6} \text{ rad/s}^2 \)

Now substituting the values:
\[ \tau = 10 \cdot \left(-\frac{1}{6}\right) = -\frac{10}{6} = -\frac{5}{3} \text{ Nm} \]
The negative sign indicates that the torque is applied in the opposite direction of the rotation.

### Step 3: Calculate the angular momentum (L) two minutes before it stops
Two minutes before it stops means we need to find the angular momentum after 8 minutes (or 480 seconds). We can calculate the angular velocity (ω) at that time using the formula:
\[ \omega = \omega_0 + \alpha t \]
Where:
- \( \omega_0 = 100 \text{ rad/s} \)
- \( \alpha = -\frac{1}{6} \text{ rad/s}^2 \)
- \( t = 480 \text{ seconds} \)

Substituting the values:
\[ \omega = 100 + \left(-\frac{1}{6}\right) \cdot 480 \]
Calculating:
\[ \omega = 100 - 80 = 20 \text{ rad/s} \]

Now, we can calculate the angular momentum (L):
\[ L = I \cdot \omega \]
Substituting the values:
\[ L = 10 \cdot 20 = 200 \text{ kg m}^2/\text{s} \]

### Summary of Results
- The torque required to stop the sphere in 10 minutes is \( -\frac{5}{3} \text{ Nm} \).
- The angular momentum of the wheel two minutes before it stops rotating is \( 200 \text{ kg m}^2/\text{s} \).