Moment of inertia corrected with relativity

[pangsiukwong]

Hi!

The moment of inertia of solid sphere is 2/5 mr^2. However, when the sphere spins at light velocity, some articles state the moment of inertia is equal to 1/2 mr^2. However, no derivation is found in those articles.

Would you suggest some articles / textbooks which show the derivation of the moment of inertia equal to 1/2 mr^2?

Patrick

 

[pervect]

I would very much like to know what your source was for the above remarks about the moment of inertia of a solid sphere.

There is some information online which would allow you to calculate the moment of inertia of a relativistic hoop (and perhaps a disk) using some assumptions about the material, i.e. a so-called hyperelastic model, at:

http://www.gregegan.net/SCIENCE/Rings/Rings.html

This is based in part upon an online discussion here at
https://www.physicsforums.com/showthread.php?t=168121

However, I am not aware of anyone textbook or article which goes through a similar derivation, nor am I aware of anyone analyzing the case of a sphere.

The above discussion is the best available information of which I am aware on this problem - it should not, however, be viewed as authoritative as that from a textbook.

The reason that a model of elasticity is needed to calculate the moment of inertia is that the stresses in the hoop (or disk, or sphere) can contribute appreciably to the moment of inertia.

If you look at the plot of angular momentum of a rotating ring vs angular frequency in the above webpage, you will see that the angular momentum actually reaches a peak, implying that the moment of inertia has dropped to zero at a high enough angular velocity.

However, a further analysis shows that when this peak is reached, the model has made non-physical assumptions. Basically, such a model represents a case of a wire that is stronger than is physically possible.

See also the important section on the limitations of the model
http://gregegan.customer.netspace.net.au/SCIENCE/Rings/Rings.html#LIMITATIONS

The model does seem to be reasonably well behaved if one is careful to avoid assumptions that make the wire "too strong". In particular, it is necessary to use values of the constants such that the speed of sound in the wire is below the speed of light in the stretched wire as well as in the unstetched wire in order to get a well-behaved model.

 

[Entropia]

Hi Patrick,

Thank you for bringing up this interesting topic! The moment of inertia is a measure of an object's resistance to rotational motion and is dependent on both the mass and distribution of mass within the object. In classical mechanics, the moment of inertia for a solid sphere is indeed 2/5 mr^2, as you mentioned.

However, when we consider the effects of relativity, the moment of inertia changes. This is because as an object approaches the speed of light, its mass increases according to the theory of special relativity. This increase in mass leads to a change in the object's moment of inertia, as the distribution of mass within the object is now different.

To derive the moment of inertia for a spinning object at relativistic speeds, we can use the Lorentz transformation equations. These equations allow us to relate the properties of an object in one reference frame to those in another reference frame moving at a constant velocity relative to the first.

I would recommend looking into textbooks on special relativity or classical mechanics that cover this topic in detail. Some examples include "Introduction to Special Relativity" by Robert Resnick and "Classical Mechanics" by Herbert Goldstein. Additionally, there are many online resources and lecture notes available that discuss this topic in depth.

I hope this helps and sparks your curiosity to further explore the fascinating world of relativity and its effects on physical quantities like the moment of inertia.