The parallel axis theorem is a fundamental concept in mechanics that relates the moment of inertia of an object about an axis to its moment of inertia about a parallel axis. It states that the moment of inertia of an object about any axis parallel to an axis passing through its center of mass is equal to the sum of the moment of inertia about the center of mass and the product of the mass of the object and the square of the distance between the two axes.

Explanation:
The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on the mass distribution of the object and the axis of rotation. The parallel axis theorem allows us to calculate the moment of inertia about an axis that does not pass through the center of mass of the object.

Here's how it works:

1. Understanding the moment of inertia: The moment of inertia of an object is mathematically defined as the sum of the products of the mass of each particle in the object and the square of its distance from the axis of rotation. In other words, it is a measure of how the mass is distributed around the axis.

2. Identifying the center of mass: The center of mass is a point that represents the average position of the mass of an object. It is the point at which the object can be balanced perfectly. For symmetrical objects, the center of mass coincides with the geometric center. However, for irregularly shaped objects, the center of mass may be located elsewhere.

3. Calculating the moment of inertia about the center of mass: The moment of inertia about the center of mass can be calculated using the standard formulas for different shapes. For example, for a thin rod rotating about an axis perpendicular to its length and passing through its center of mass, the moment of inertia is given by (1/12) * m * L^2, where m is the mass of the rod and L is its length.

4. Calculating the moment of inertia about a parallel axis: To calculate the moment of inertia about a parallel axis, we need to consider the additional contribution due to the distance between the two axes. The parallel axis theorem states that the moment of inertia about the parallel axis is equal to the moment of inertia about the center of mass plus the product of the mass of the object and the square of the distance between the two axes. Mathematically, it can be written as I_parallel = I_cm + m * d^2, where I_parallel is the moment of inertia about the parallel axis, I_cm is the moment of inertia about the center of mass, m is the mass of the object, and d is the distance between the two axes.

5. Using the square of the distance: The parallel axis theorem uses the square of the distance between the two axes because the moment of inertia is a second-order property. The square of the distance ensures that the contribution to the moment of inertia is proportional to the area or mass distribution of the object in relation to the distance from the axis of rotation.

In conclusion, the parallel axis theorem uses the square of the distance to calculate the moment of inertia about a parallel axis. This theorem is a powerful tool in mechanics and is used to analyze the rotational motion of objects with complex mass distributions.