Understanding the Maximum Moment of Inertia: Why It Is Always Less Than or Equal to ( MR^2 )

In physics, moment of inertia is a measure of an object's resistance to changes in its rotational motion. If we consider a particle at a distance ( r ) from an axis of rotation, the moment of inertia is given by ( mr^2 ). When dealing with a rigid body, the moment of inertia can be expressed as the sum of the individual moments of inertia of each element of mass that makes up the body.

Defining the Moment of Inertia

The moment of inertia I of a system of particles can be written as:

I m1r12 m2r22 ... miri2

Here, mi represents the mass of the ith particle and ri is its distance from the axis of rotation. By definition, the moment of inertia is maximized when all the mass is at the greatest possible distance from the axis.

Why Can't the Moment of Inertia Exceed ( MR^2 )?

The moment of inertia of a rigid body confined to a cylindrical volume with radius ( R ) can be at most ( MR^2 ). This is because the total moment of inertia is the sum of the moments of inertia of all the individual elements of mass. If we assume that the mass ( M ) is distributed such that the farthest distance ( r ) from the axis is ( R ), then the maximum moment of inertia is achieved when all the mass is at this distance ( R ).

The formula for the moment of inertia in this case becomes:

I Mr2 MR2

This is true for any rigid body with all its mass confined within a cylindrical volume of radius ( R ). If the mass is distributed closer to the axis, the moment of inertia will be less than ( MR^2 ).

Maximizing Moment of Inertia: The Hoop Example

The shape that maximizes the moment of inertia for a given mass and radius is a hoop (or a circular ring). In a hoop, all the mass is at the maximum distance ( R ) from the axis. Therefore, the moment of inertia for such a shape is:

I MR2

For any other shape, the moment of inertia will be less than ( MR^2 ) because the mass distribution is more concentrated towards the center, resulting in smaller values of ( r^2 ) for a larger number of mass elements.

Further Considerations on Mass Distribution

The more matter that is farther from the axis of rotation, the higher the moment of inertia will be. Among the various shapes, the one with the most mass away from the rotation axis is a hoop. Other shapes, whether they are disks, spheres, or other polyhedrons, will either have a larger ( R ) or more mass closer to the center, resulting in a lower moment of inertia.

To summarize, the maximum moment of inertia of a body with mass ( M ) and radius ( R ) is always ( MR^2 ). This occurs when the mass is distributed such that all the mass is at the greatest possible distance from the axis of rotation.