Let `I_(A)` and `I_(B)` be moments of inertia of a body about two axes `A` and `B` respectively. The axis `A` passes through the centre of mass of the body but `B` does not. Then

Options

A. `I_AltI_B`
B. `If I_AltI_B` the axes pareparallel
C. `If the axes are parallel I_AltI_B`
D. `If the axes are not parallel I_AgeI_B`

Answer

To solve the problem regarding the moments of inertia \( I_A \) and \( I_B \) about two different axes \( A \) and \( B \), we can follow these steps:

### Step 1: Understand the Concept of Moment of Inertia
The moment of inertia of a body is a measure of how difficult it is to change its rotational motion about a given axis. It depends on the mass distribution of the body relative to the axis of rotation.

### Step 2: Identify the Axes
- Axis \( A \) passes through the center of mass (CM) of the body.
- Axis \( B \) does not pass through the center of mass.

### Step 3: Apply the Parallel Axis Theorem
The Parallel Axis Theorem states that if you know the moment of inertia \( I_{CM} \) about an axis through the center of mass, you can find the moment of inertia about any parallel axis using the formula:

\[ I_B = I_{CM} + Md^2 \]

where:
- \( I_B \) is the moment of inertia about axis \( B \),
- \( I_{CM} \) is the moment of inertia about the center of mass (which is \( I_A \)),
- \( M \) is the total mass of the body,
- \( d \) is the distance between the two axes.

### Step 4: Calculate Moments of Inertia
For axis \( A \) (through the center of mass):
\[ I_A = I_{CM} \]

For axis \( B \) (not through the center of mass):
\[ I_B = I_A + Md^2 \]

### Step 5: Compare \( I_A \) and \( I_B \)
Since \( d \) is a positive distance (as \( B \) does not pass through the center of mass), we can conclude that:
\[ I_B = I_A + Md^2 > I_A \]

This implies that:
\[ I_A < I_B \]

### Conclusion
Thus, we have shown that the moment of inertia about the axis passing through the center of mass \( I_A \) is less than the moment of inertia about the axis \( B \) that does not pass through the center of mass.

### Final Answer
\[ I_A < I_B \]