Animated Solution for Physics - Rotational Motion: Four identical solid spheres each of mass $m$ and radius $a$ are placed with their centres on the four corners of a square of side $b$. The moment of inertia of the system about one side of square, where the axis of rotation is parallel to the plane of the square is

Imagine you are tasked with spinning a heavy, rigid structure. The effort required to get it spinning—or to stop it once it's moving—depends entirely on its Moment of Inertia. This physical quantity is the rotational analogue of mass. However, unlike mass, which is a fixed property of an object, the moment of inertia depends heavily on how that mass is distributed relative to the axis of rotation.

In this fascinating problem, we are presented with a highly symmetrical system: four identical solid spheres, each of mass $m$ and radius $a$, positioned perfectly at the corners of a square of side $b$. Our mission is to calculate the total moment of inertia of this entire system about an axis that passes exactly through one of the sides of the square.

The Intrinsic Inertia

Before we look at the system as a whole, we must understand the individual components. Every solid sphere possesses an intrinsic resistance to rotation about its own central diameter. For a uniform solid sphere, this is a standard, derived result:

This tells us that even if a sphere is spinning perfectly in place, it carries this baseline rotational inertia.

The Parallel Axis Theorem

When an object rotates about an axis that does not pass through its center of mass, it sweeps out a larger circular path, effectively increasing its rotational inertia. To calculate this new inertia, we employ one of the most powerful tools in classical mechanics: the Parallel Axis Theorem.

This theorem elegantly states that the total moment of inertia ($I$) is equal to the object's intrinsic inertia about its center of mass ($I_{\text{cm}}$) plus an additional term ($m{d}^2$) that accounts for the mass ($m$) being shifted by a perpendicular distance ($d$) from the new axis.

Breaking Down the System

Let's choose the top side of the square as our axis of rotation. This axis connects the centers of the two top spheres (let's call them A and B).

For Spheres A and B: Because the axis of rotation passes directly through their centers, their perpendicular distance $d$ from the axis is exactly zero ($d=0$). Therefore, the parallel axis theorem simply returns their intrinsic inertia. Since there are two such spheres, their combined contribution is:

For Spheres C and D: These two spheres sit at the bottom corners of the square. Their centers are separated from our chosen axis by a perpendicular distance equal to the side of the square, so $d=b$. Applying the parallel axis theorem to each of these spheres yields:

Since there are two identical spheres at this distance, their combined contribution is:

The Final Synthesis

Because moment of inertia is a scalar quantity, the total moment of inertia of a composite system is simply the algebraic sum of the moments of inertia of its individual parts. We add the contributions from the top spheres and the bottom spheres:

Combining the like terms, we arrive at our final, elegant expression:

This result beautifully encapsulates both the intrinsic geometry of the spheres (the $a^2$ term) and the macroscopic geometry of the square arrangement (the $b^2$ term).