A rigid body of mass m rotates with the angular velocity __ about an axis at a distance 'a' from the centre of mass G. The radius of gyration about G is K. Then kinetic energy of rotation of the body about new parallel axis is

Options

A. `(1)/(2)mK^(2)omega^(2)`
B. `(1)/(2)ma^(2)omega^(2)`
C. `(1)/(2)m(a^(2)+K^(2))omega^(2)`
D. `(1)/(2)m(a+K^(2))omega^(2)`

Explanation

To find the kinetic energy of rotation of a rigid body about a new parallel axis, we can follow these steps:

### Step 1: Understand the Given Information
We have a rigid body of mass \( m \) rotating with an angular velocity \( \omega \) about an axis at a distance \( a \) from its center of mass \( G \). The radius of gyration about \( G \) is given as \( K \).

### Step 2: Moment of Inertia about the Center of Mass
The moment of inertia \( I_{G} \) of the body about the center of mass \( G \) can be expressed using the radius of gyration \( K \):
\[ I_{G} = m K^2 \]

### Step 3: Use the Parallel Axis Theorem
To find the moment of inertia \( I \) about the new axis that is parallel to the one through the center of mass and at a distance \( a \), we use the parallel axis theorem:
\[ I = I_{G} + m a^2 \]
Substituting for \( I_{G} \):
\[ I = m K^2 + m a^2 \]
This simplifies to:
\[ I = m (K^2 + a^2) \]

### Step 4: Calculate the Kinetic Energy of Rotation
The kinetic energy \( K.E. \) of rotation about the new axis can be calculated using the formula:
\[ K.E. = \frac{1}{2} I \omega^2 \]
Substituting the expression for \( I \):
\[ K.E. = \frac{1}{2} (m (K^2 + a^2)) \omega^2 \]
Thus, the final expression for the kinetic energy of rotation about the new axis is:
\[ K.E. = \frac{1}{2} m (K^2 + a^2) \omega^2 \]

### Summary
The kinetic energy of rotation of the body about the new parallel axis is:
\[ K.E. = \frac{1}{2} m (K^2 + a^2) \omega^2 \]

---