Sorry the solution is not understandable...my doubt is perpendicular axes theorem is applied to Iz = Ix + Iy.....

The Perpendicular Axes Theorem is a fundamental concept in physics and engineering, particularly in the study of moments of inertia. It states that if you have a flat, planar object, the moment of inertia about an axis perpendicular to the plane of the object (let's call it the z-axis) can be calculated by summing the moments of inertia about two perpendicular axes (the x-axis and the y-axis) that lie in the same plane as the object. This can be expressed mathematically as Iz = Ix + Iy.

Understanding the Components of the Theorem

To grasp this theorem, it’s essential to break down the components:

• Moment of Inertia (I): This is a measure of an object's resistance to rotational motion about an axis. It depends on the mass distribution relative to that axis.
• Axes: The x and y axes are in the plane of the object, while the z-axis is perpendicular to that plane.

Visualizing the Concept

Imagine a flat rectangular sheet of metal lying on a table. If you want to find out how difficult it is to rotate this sheet around an axis that goes straight up through the center (the z-axis), you can instead calculate how difficult it is to rotate it around two axes that lie flat on the table (the x and y axes).

For instance, if you know the moment of inertia about the x-axis (Ix) and the moment of inertia about the y-axis (Iy), you can easily find the moment of inertia about the z-axis (Iz) by simply adding these two values together:

Iz = Ix + Iy

Why This Works

The reason this theorem holds true is rooted in the way rotational motion works. When you rotate an object, the distribution of mass relative to the axis of rotation determines how much torque is needed to achieve that rotation. By considering the contributions of the mass distribution along both the x and y axes, you can effectively capture the total resistance to rotation about the z-axis.

Example for Clarity

Let’s say you have a thin, uniform rectangular sheet with a width of 2 meters and a height of 1 meter. The mass of the sheet is 4 kg. To find the moments of inertia:

• For the x-axis (horizontal axis through the center), you can use the formula for a rectangle: Ix = (1/12) * m * h^2, where h is the height.
• For the y-axis (vertical axis through the center), the formula is similar: Iy = (1/12) * m * w^2, where w is the width.

Plugging in the values:

• Ix = (1/12) * 4 kg * (1 m)^2 = (1/12) * 4 = 0.333 kg·m²
• Iy = (1/12) * 4 kg * (2 m)^2 = (1/12) * 16 = 1.333 kg·m²

Now, applying the Perpendicular Axes Theorem:

Iz = Ix + Iy = 0.333 kg·m² + 1.333 kg·m² = 1.666 kg·m²

Applications of the Theorem

This theorem is particularly useful in mechanical engineering and physics when analyzing the rotational dynamics of objects. It simplifies calculations significantly, especially when dealing with complex shapes or systems where direct calculation of Iz might be cumbersome.

In summary, the Perpendicular Axes Theorem provides a straightforward method to determine the moment of inertia about an axis perpendicular to a plane by leveraging the moments of inertia about two axes in that plane. This principle is not only mathematically elegant but also practically invaluable in various fields of study. If you have any more questions or need further clarification, feel free to ask!