I "Moment of Inertia" in Virial Theorem

[throneoo]

Moment of inertia is supposed to be defined with respect to a rotational axis such that for a system of point masses, I=∑m_ir_i² where r_i 's are the perpendicular distances of the particles from the axis.

However, in some derivations of the virial theorem (like the one on wiki), the so-called "scalar" moment of inertia, the r_i 's are taken to be the magnitude of the position vectors of those particles with respect to the origin without reference to any axis. My question is, does it still have the same physical significance as its ordinary counterpart? This quantity at most indicates the overall separation of the particles from the origin

 

[Ken G]

Moment of inertia is supposed to be defined with respect to a rotational axis such that for a system of point masses, I=∑m_ir_i² where r_i 's are the perpendicular distances of the particles from the axis.

Actually, that's not the general definition of a moment of inertia. That only applies for rigid bodies that have enough symmetry to be rotating around a fixed axis, more general rigid bodies have only a moment of inertia tensor and can have their axis of rotation wobble. Even when we choose axes that make the moment of inertia tensor diagonal, it only means that there will be different moments of inertia of the form you mention around each of those axes (and rotation around the axis with the middle-sized moment of inertia will not be stable, and will wobble). The most general definition of the moment of inertia of a rigid body can be found here: http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node64.html

However, in some derivations of the virial theorem (like the one on wiki), the so-called "scalar" moment of inertia, the r_i 's are taken to be the magnitude of the position vectors of those particles with respect to the origin without reference to any axis.

The virial theorem is usually used on collections of particles that are not rigid, and the appearance of a quantity that in some superficial ways resembles a moment of inertia is just a coincidence. It's not the moment of inertia.

 

[throneoo]

Actually, that's not the general definition of a moment of inertia. That only applies for rigid bodies that have enough symmetry to be rotating around a fixed axis, more general rigid bodies have only a moment of inertia tensor and can have their axis of rotation wobble. Even when we choose axes that make the moment of inertia tensor diagonal, it only means that there will be different moments of inertia of the form you mention around each of those axes (and rotation around the axis with the middle-sized moment of inertia will not be stable, and will wobble). The most general definition of the moment of inertia of a rigid body can be found here: http://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node64.html
The virial theorem is usually used on collections of particles that are not rigid, and the appearance of a quantity that in some superficial ways resembles a moment of inertia is just a coincidence. It's not the moment of inertia.

after readig ur link i realized talking about the moment of inertia of a non rigid collection of particles does not make much sense lol.
thanks.