If a body completes one revolution in `pi` sec then the moment of inertia would be

Options

A. Equal to rotational kinetic energy
B. Double of rotational kinetic energy
C. Half of rotational kinetic energy
D. Four times the rotational kinetic energy

Answer

To solve the problem of finding the moment of inertia of a body that completes one revolution in \( \pi \) seconds, we can follow these steps:

### Step 1: Understand the relationship between angular velocity and time
Given that the body completes one revolution in \( \pi \) seconds, we can determine the angular velocity (\( \omega \)) of the body. One complete revolution corresponds to an angle of \( 2\pi \) radians.

\[ \omega = \frac{\text{Total angle in radians}}{\text{Time in seconds}} = \frac{2\pi \text{ radians}}{\pi \text{ seconds}} = 2 \text{ radians/second} \]

### Step 2: Write the formula for rotational kinetic energy
The rotational kinetic energy (\( K \)) of a body is given by the formula:

\[ K = \frac{1}{2} I \omega^2 \]

where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

### Step 3: Substitute the value of \( \omega \) into the kinetic energy formula
Now, substituting \( \omega = 2 \text{ radians/second} \) into the kinetic energy formula:

\[ K = \frac{1}{2} I (2)^2 = \frac{1}{2} I \cdot 4 = 2I \]

### Step 4: Rearranging the equation to find the moment of inertia
To find the moment of inertia \( I \), we can rearrange the equation:

\[ I = \frac{K}{2} \]

### Step 5: Conclusion
Thus, the moment of inertia \( I \) is half of the rotational kinetic energy \( K \):

\[ I = \frac{K}{2} \]

### Final Answer
The moment of inertia of the body is \( \frac{K}{2} \), where \( K \) is the rotational kinetic energy of the body.

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