Animated Solution for Physics - Rotational Motion: Moment of inertia (MI) of four bodies, having same mass and radius, are reported as $I_1=$ MI of thin circular ring about its diameter, $I_2=$ MI of circular disk about an axis perpendicular to the disk and going through the centre, $I_3=$ MI of solid cylinder about its axis and $I_4=$ MI of solid sphere about its diameter. Then,

Ring, Disk, Cylinder, and Sphere

Imagine you are tasked with spinning four different objects: a thin circular ring, a solid disk, a solid cylinder, and a solid sphere. They all weigh exactly the same (mass $M$) and have the exact same outer radius ($R$). The question is, which one is the easiest to spin, and which ones share the exact same resistance to rotational motion? Let's break down their moments of inertia step by step.

First, let's look at the thin circular ring. The problem specifies that it is rotating about its diameter. We know that the moment of inertia of a ring about an axis perpendicular to its plane passing through the center is $M{R}^2$. By applying the perpendicular axis theorem ($I_z=I_x+I_y$), the moment of inertia about any diameter is exactly half of that.

Next, we have the circular disk. The axis of rotation here is perpendicular to the disk and passes right through its center. This is a standard derivation in rotational mechanics. The mass is distributed evenly from the center to the edge, resulting in:

Now, visualize the solid cylinder rotating about its longitudinal axis. You can think of a solid cylinder as nothing more than a tall stack of many identical thin disks. Since the distance of every mass element from the central axis is distributed in the exact same way as it is for a single disk, its moment of inertia formula remains identical to the disk!

Finally, we arrive at the solid sphere rotating about its diameter. Think about the shape of a sphere compared to a cylinder of the same radius. The sphere tapers off at the top and bottom poles. This means that, on average, more of the sphere's mass is concentrated closer to the central axis of rotation compared to the uniform cylinder. Because the mass is closer to the axis, its resistance to rotation (moment of inertia) is slightly less.

Let's put all these values side by side and compare their coefficients:

$I_1=0.5M{R}^2$ $I_2=0.5M{R}^2$ $I_3=0.5M{R}^2$ $I_4=0.4M{R}^2$

The conclusion is crystal clear. The ring (about its diameter), the disk (about its center), and the solid cylinder (about its axis) all share the exact same moment of inertia. However, the solid sphere has a smaller moment of inertia because its mass is more centrally concentrated.

Therefore, the final relationship is:

This perfectly matches option (d).