The theorem of perpendicular axes is applicable for

Options

A. only planar bodies
B. only regular shaped bodies
C. only three dimensional bodies
D. None of the above

Answer

To determine the applicability of the theorem of perpendicular axes, we need to understand the conditions under which this theorem holds true. The theorem states that for a planar body, the moment of inertia about an axis perpendicular to the plane (Iz) is equal to the sum of the moments of inertia about two perpendicular axes lying in the plane (Ix and Iy).

### Step-by-Step Solution:

1. **Understanding the Theorem**: The theorem of perpendicular axes states that for a planar body, the moment of inertia about an axis perpendicular to the plane (Iz) is equal to the sum of the moments of inertia about two perpendicular axes in the plane (Ix + Iy). This can be mathematically expressed as:
\[ I_z = I_x + I_y \]

2. **Identifying Planar Bodies**: A planar body is a two-dimensional object that lies entirely in a single plane. Examples include shapes like rectangles, circles, and triangles. The theorem is specifically applicable to these kinds of bodies.

3. **Exclusion of 3D Bodies**: The theorem does not apply to three-dimensional bodies (like spheres or cubes) because their moment of inertia cannot be simply derived from the moments of inertia of two axes in a plane. For 3D bodies, the distribution of mass in three dimensions complicates the calculation of moment of inertia.

4. **Conclusion**: Based on the above understanding, we conclude that the theorem of perpendicular axes is applicable only for planar bodies.

### Final Answer:
The theorem of perpendicular axes is applicable for **planar bodies**.

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