# A Contradiction in Moment of Inertia Formulae by reductio ad absurdum?

## Main Question or Discussion Point

In physics the moment of inertia of a thin rod which rotates around an axis through its center of mass is :

I cen = 1/12 m L sq (1)

Where: m is the mass of the rod, L is the length of the rod.

The moment of inertia of a thin rod which rotates around an axis which is at one end of the rod is:

I end = 1/3 m L sq (2)

However, this seems to lead to a paradox as follows: Imagine a rod of length L which rotates around an axis through its center of mass. Imagine the rod is split into 2 equal half-lengths with each half-length equal to L/2.
Thus, each half-length rod is rotating around the axis at one end. The moment of inertia of each half-length is, by equation (2):

I half-length end = 1/3 m (1/2 L) sq (3)

or

I half_length end = 1/12 m L sq (4)

But this is the same as equation (1). How can this be? If you computed the total moment of inertia of both half-lengths, it would equal:

I half-length end * 2 = 1/6 m L sq (5)

Which would leads to the impossible situation of (5) being greater than (1). Hence, we would have two different derivations of the moment of inertia of the same rod that give two different answers.

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