A body rotating with angular velocity `omega` stops after rotating for 2 sec. The torque necessary for this is?

Options

A. `(Iomega)/2`
B. `2Iomega`
C. `Iomega`
D. `(Iomega^2)/2`

Answer

To solve the problem of finding the torque necessary to stop a body rotating with an angular velocity \( \omega \) after 2 seconds, we can follow these steps:

### Step 1: Understand the problem
We know that the body is rotating with an initial angular velocity \( \omega \) and needs to come to a stop (final angular velocity \( \omega_f = 0 \)) in a time duration of \( t = 2 \) seconds.

### Step 2: Use the angular motion equation
We can use the first equation of angular motion, which relates initial angular velocity, final angular velocity, angular acceleration, and time:
\[ \omega_f = \omega_i + \alpha t \]
Where:
- \( \omega_f \) = final angular velocity (0 in this case)
- \( \omega_i \) = initial angular velocity (\( \omega \))
- \( \alpha \) = angular acceleration
- \( t \) = time (2 seconds)

### Step 3: Substitute the known values
Substituting the known values into the equation:
\[ 0 = \omega + \alpha \cdot 2 \]

### Step 4: Solve for angular acceleration \( \alpha \)
Rearranging the equation to solve for \( \alpha \):
\[ \alpha \cdot 2 = -\omega \]
\[ \alpha = -\frac{\omega}{2} \]
The negative sign indicates that the angular acceleration is in the opposite direction to the initial angular velocity.

### Step 5: Relate torque to angular acceleration
The torque \( \tau \) required to produce an angular acceleration \( \alpha \) is given by the equation:
\[ \tau = I \alpha \]
Where \( I \) is the moment of inertia of the body.

### Step 6: Substitute \( \alpha \) into the torque equation
Substituting the expression for \( \alpha \) into the torque equation:
\[ \tau = I \left(-\frac{\omega}{2}\right) \]
Thus, the magnitude of the torque required to stop the body is:
\[ \tau = \frac{I \omega}{2} \]

### Final Answer
The torque necessary for the body to stop after 2 seconds is:
\[ \tau = \frac{I \omega}{2} \]