I have previously wondered about why mass is additive; i.e. the translational kinematics of some bodies can be determined by considering them as a single body with a total mass equal to the sum of the individual masses, acting at a point which we call the centre of mass. I thought that this probably had to do with linearity of the momentum operator, but then I started thinking about what it means for the momentum operator to be linear, physically. I never really found a satisfactory explanation, and I just forgot about the problem.
However now the problem has been rekindled considering the moment of inertia. I find it even more perplexing that the moment of inertia of some compound object about a given axis can be summed by finding the sum of the individual moments of inertia! This is particularly puzzling for me because the moment of inertia is proportional to the distance squared (although perhaps this has nothing to do with the problem!)
I would be very grateful if someone had some explanation as to why these quantities are additive- particularly the moment of inertia!
Best Answer
That mass is additive is an empirical fact which we use without question in constructing our physical and mathematical models of the universe. We could imagine that if the universe had different Laws Of Nature then bringing two bodies together might increase their combined mass above the sum of their separate masses, just as bringing two charges together increases the force between them.
An ultimate Theory of Everything might "explain" this property in terms of some abstract concepts, but this will only "prove" that that theory is consistent with our observations.
(I have ignored what happens in Special Relativity because you have used the tag for Newtonian Mechanics.)
The additive property of the moment of inertia is inherent its definition : $$ I = \Sigma m_i r_i^2$$ Any number of sub-sets of particles could be summed separately and would still give the same total moment of inertia.
(I note that you are asking about moments about the same "given" axis. There is no simple addition if the axes are not the same.)
Yes, this fact is related to the additivity of momentum, because angular momentum is defined as the moment of momentum :
$$L=\Sigma m_iv_ir_i=\Sigma (m_i r_i^2)\frac{v_i}{r_i} =I\omega$$
where $\frac{v_i}{r_i}$ is the same for all particles in a rigid body.