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When we talk about the moment of inertia, one key factor to consider is the location of the rotation axis. This axis is an imaginary line around which an object spins or rotates. It can be compared to a pivot point that determines the manner in which the body's parts contribute to the moment of inertia.
The distribution of mass relative to this axis greatly impacts how much effort is required to rotate the object. In the problem you provided, the axis of rotation is crucial because it changes when the shape of the object changes—from a sphere to a disk. For a sphere, the axis passes through its center; while for a disk, the axis is perpendicular to and passes through the center of its flat surface.
The change in the shape of the object and the associated change in the rotation axis are what lead to different moments of inertia. This change shows how intricately the axis and geometry of objects govern rotational dynamics.

A solid sphere is a three-dimensional round object, like a ball. Its moment of inertia depends on its size and mass distribution. In mathematical terms, for a solid sphere, the moment of inertia about an axis through its center is calculated using the formula: \(\frac{2}{5}MR^2\). Here, \(M\) represents the mass of the sphere, and \(R\) is its radius.
This formula tells us that a significant portion of a sphere's mass is distributed around its center. The smaller the radius, the easier it is to rotate the sphere around this axis due to the alignment of mass towards the center. This centralized mass distribution plays a big role in defining the sphere's moment of inertia. It's why we calculate \(I_1\) using this specific formula in the provided exercise.

A circular disk can be visualized like a flat plate or a Frisbee, and it is this shape that the sphere in the exercise is flattened into, keeping the same radius. When calculating the moment of inertia for such a disk, we use the formula: \(\frac{1}{2}MR^2\). In this formula, \(M\) is the mass of the disk, and \(R\) is the radius.
The disk's mass distribution around its central axis significantly affects its rotational properties. Unlike a sphere, a disk has much of its mass spread out from the center, making the effort needed to rotate it different. This new distribution helps explain why the moment of inertia of the disk, \(I_2\), results in a different value compared to the sphere, specifically \(\frac{5}{4}I_1\), as calculated using the problem's solution.

Mass distribution refers to how mass is spread over the volume of an object. This factor is crucial in determining the moment of inertia. The more the mass is spread away from the axis of rotation, the larger the moment of inertia will be. This principle explains why different shapes with the same mass and size can have vastly different rotational behaviors.
In the exercise, when the spherical clay is flattened into a disk, its mass distribution changes. Although the mass stays constant, the shape and how that mass is spread out alters the inertia characteristics. The disk has its mass more evenly spread in a flat plane, which increases its moment of inertia compared to when it was a sphere. Understanding this link between shape, mass distribution, and inertia helps explain the nature of rotational dynamics in different objects—providing a solid foundation for understanding physics principles in rotational motion.