Animated Solution for Physics - Rotational Motion: Moment of inertia of a circular wire of mass $M$ and radius $R$ about its diameter is

The moment of inertia is a beautiful concept that tells us how much an object resists rotational acceleration. When dealing with highly symmetric objects like a circular wire (or a ring), calculating the moment of inertia becomes an elegant exercise in geometry and physics. Let's dive into how we can find the moment of inertia of a ring about its diameter.

Imagine a uniform circular wire of mass $M$ and radius $R$. If we were to spin this ring like a top around an axis passing straight through its center and perpendicular to its plane, every single tiny mass element $dm$ of the ring would be at the exact same distance $R$ from the axis.

Because the moment of inertia is the sum of $dm\cdot r^2$, and $r$ is a constant $R$ for all elements, the total moment of inertia about this central $z$-axis is simply:

But what if we want to flip the ring like a coin? That means we need to find the moment of inertia about an axis that lies in the plane of the ring—its diameter.

To solve this, we bring in a powerful tool: the Perpendicular Axes Theorem. This theorem is a lifesaver for 2D planar objects. It states that the moment of inertia about an axis perpendicular to the plane ($I_z$) is equal to the sum of the moments of inertia about two mutually perpendicular axes lying in the plane ($I_x$ and $I_y$), provided all three axes intersect at the same point.

Mathematically, it is written as:

Let's set our $x$-axis and $y$-axis as two perpendicular diameters of the ring. Because the ring is perfectly symmetric, it doesn't matter which diameter we choose; the resistance to rotation will be exactly the same. Therefore, the moment of inertia about the $x$-axis must equal the moment of inertia about the $y$-axis. Let's call this common value $I$.

Now, we just plug our knowns into the theorem. We know $I_z=M{R}^2$, and we know $I_x+I_y=I+I=2I$.

Substituting these into the theorem gives:

To find the moment of inertia about the diameter ($I$), we simply divide by 2:

And there we have it! The moment of inertia of a circular wire about its diameter is exactly half of its moment of inertia about its central perpendicular axis. This is a fundamental result that frequently appears in rotational mechanics, and understanding its derivation through symmetry and the perpendicular axes theorem is key to mastering rigid body dynamics.