A torque T acts on a body of moment of inertia l rotating with angular speed `omega`. It will be stopped just after time

Options

A. `(lT)/(omega)`
B. `(l omega)/(T)`
C. `(T omega)/(l)`
D. `l omega T`

Answer

To solve the problem step by step, we will analyze the relationship between torque, moment of inertia, angular acceleration, and angular velocity.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- A torque \( T \) acts on a body.
- The moment of inertia of the body is \( I \).
- The initial angular speed of the body is \( \omega \).
- The body comes to rest after some time \( t \).

2. **Relating Torque and Angular Acceleration**:
- The relationship between torque \( T \), moment of inertia \( I \), and angular acceleration \( \alpha \) is given by the equation:
\[ T = I \alpha \]
- Since the torque is acting in the opposite direction to the angular velocity (as the body is coming to rest), we can express angular acceleration as:
\[ \alpha = -\frac{T}{I} \]

3. **Using Kinematic Equation for Angular Motion**:
- The final angular velocity \( \omega_f \) when the body comes to rest is 0. We can use the kinematic equation for angular motion:
\[ \omega_f = \omega_i + \alpha t \]
- Substituting \( \omega_f = 0 \) and \( \omega_i = \omega \):
\[ 0 = \omega - \alpha t \]
- Rearranging gives:
\[ \alpha t = \omega \]
- Thus,
\[ t = \frac{\omega}{\alpha} \]

4. **Substituting for Angular Acceleration**:
- We already found that \( \alpha = -\frac{T}{I} \). Substituting this into the equation for time:
\[ t = \frac{\omega}{-\frac{T}{I}} = -\frac{I \omega}{T} \]
- Since time cannot be negative, we take the absolute value:
\[ t = \frac{I \omega}{T} \]

5. **Final Result**:
- The time taken for the body to come to rest is:
\[ t = \frac{I \omega}{T} \]

### Conclusion:
The correct answer to the question is \( t = \frac{I \omega}{T} \).