Why should the value of moment of inertia be relatively close to one another for two different mass but on the same body?

The value of moment of inertia depends not only on the mass of an object but also on its distribution of mass around its axis of rotation. It is a measure of an object's resistance to rotational motion.

When two different masses are placed on the same body, the distance of those masses from the axis of rotation determines their contribution to the moment of inertia. If these distances are relatively close to each other, then the moment of inertia for both masses will be similar.

To understand why this happens, we can use the formula for moment of inertia of a point mass:

I = mr^2

where I is the moment of inertia, m is the mass, and r is the perpendicular distance from the axis of rotation to the mass.

When the distances (r) of two different masses are close, they contribute similar values to the overall moment of inertia. This means that the distribution of mass around the axis of rotation is more uniform, leading to a relatively consistent value of moment of inertia.

In practical terms, when the masses are distributed evenly or are concentrated close to the axis of rotation, the moment of inertia for different masses on the same body tends to be close to one another. This is particularly important in systems involving rotational motion, such as wheels, gears, or pendulums, where having a consistent moment of inertia helps in predicting and analyzing their behavior.