The parallel axis theorem can be applied to

Options

A. any two parallel axes
B. any two parallel-axes of which one must lie within the body
C. any two parallel-axes of which one must pass through the centre of mass of the body.
D. any two parallel axes lying in the plane of the body

Answer

### Step-by-Step Solution:

1. **Understanding the Parallel Axis Theorem**: The parallel axis theorem is a principle in rotational mechanics that allows us to calculate the moment of inertia of a body about any axis parallel to an axis through its center of mass.

2. **Identifying the Axes**: Let's denote two axes:
- Axis 1 (ICM): This axis passes through the center of mass (CM) of the body.
- Axis 2 (Ip): This axis is parallel to Axis 1 and is located at a distance 'd' from it.

3. **Moment of Inertia Calculation**: According to the parallel axis theorem, the moment of inertia (Ip) about Axis 2 can be expressed as:
\[ I_p = I_{cm} + m \cdot d^2 \]
where:
- \(I_{cm}\) is the moment of inertia about the center of mass axis (Axis 1).
- \(m\) is the mass of the body.
- \(d\) is the distance between the two parallel axes.

4. **Application Conditions**: The theorem can be applied to any rigid body, whether it is two-dimensional or three-dimensional, as long as:
- One of the axes must pass through the center of mass.
- The two axes must be parallel.

5. **Conclusion**: Therefore, the parallel axis theorem can be applied to any two parallel axes where one axis is through the center of mass of the body.

### Final Answer:
The parallel axis theorem can be applied to any two parallel axes, provided that one of them passes through the center of mass of the body.

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