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

[e2m2a]

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.

 

[Rancho]

There is no paradox here...
The procedure is right but, the only thing is when u split the rod into 2 halves, u must take the mass of each part as "m/2" and not "m".

 

[e2m2a]

There is no paradox here...
The procedure is right but, the only thing is when u split the rod into 2 halves, u must take the mass of each part as "m/2" and not "m".

You're right. Thanks for pointing out the error.

 

[Rancho]

anytime..when I'm ol :)

 

[CellCoree]

I would first review the derivation of the moment of inertia formulae for a thin rod rotating around an axis through its center of mass and an axis at one end. I would also check for any assumptions or simplifications that were made in the derivation process.

If the derivations are correct and there are no errors, then this contradiction could possibly be resolved by considering the distribution of mass along the rod. The moment of inertia formulae assume a uniform mass distribution, but in reality, the mass of the rod may not be evenly distributed.

Additionally, this paradox could also be explained by the fact that the moment of inertia is a measure of an object's resistance to rotational motion. Therefore, the moment of inertia of the two half-length rods rotating around the axis at one end would be different than the moment of inertia of the full rod rotating around an axis through its center of mass. This is because the two half-length rods have different distributions of mass and therefore different resistances to rotational motion.

In conclusion, while this apparent contradiction may seem puzzling at first, it can be resolved by considering the assumptions and limitations of the moment of inertia formulae and the distribution of mass along the rod. It is important for scientists to continuously review and refine their theories and equations to ensure accuracy and avoid contradictions.