A square plate is kept in YZ plane. Then according to perpendicular axis theorem

Options

A. `I_Z=I_X+I_Y`
B. `I_X=I_Y+I_Z`
C. `I_Y=I_X+I_Z`
D. All of the above

Answer

To solve the problem regarding the perpendicular axis theorem for a square plate kept in the YZ plane, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have a square plate lying in the YZ plane. The axes are defined as follows: Y-axis is vertical, and Z-axis is horizontal. The X-axis is perpendicular to the YZ plane.

2. **Defining the Moment of Inertia**:
- According to the perpendicular axis theorem, for a planar body (like our square plate), the moment of inertia about an axis perpendicular to the plane (X-axis) is equal to the sum of the moments of inertia about the two axes in the plane (Y-axis and Z-axis).
- Mathematically, this can be expressed as:
\[ I_x = I_y + I_z \]
where:
- \(I_x\) is the moment of inertia about the X-axis,
- \(I_y\) is the moment of inertia about the Y-axis,
- \(I_z\) is the moment of inertia about the Z-axis.

3. **Calculating the Moments of Inertia**:
- For a square plate of mass \(m\) and side length \(a\):
- The moment of inertia about the Y-axis (\(I_y\)) can be calculated as:
\[ I_y = \frac{1}{12} m a^2 \]
- The moment of inertia about the Z-axis (\(I_z\)) can be calculated similarly:
\[ I_z = \frac{1}{12} m a^2 \]

4. **Applying the Perpendicular Axis Theorem**:
- Substitute the values of \(I_y\) and \(I_z\) into the equation:
\[ I_x = I_y + I_z = \frac{1}{12} m a^2 + \frac{1}{12} m a^2 = \frac{1}{6} m a^2 \]

5. **Conclusion**:
- Therefore, the relation according to the perpendicular axis theorem for the square plate in the YZ plane is:
\[ I_x = I_y + I_z \]
- This confirms that the moment of inertia about the X-axis is equal to the sum of the moments of inertia about the Y and Z axes.