Why are mass and the moment of inertia additive?

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!

• Hmmm ... moment of inertia is only additive if you either (a) have the axes for each body co-linear or (b) first re-compute the moment for one or both bodies to achieve (a). – dmckee --- ex-moderator kitten Feb 26 '17 at 18:24
• mass is additive only in classical physics. In special relativity it is not, it is the "length" of the four dimensional pseudovector ( momentum vector, energy). – anna v Feb 26 '17 at 18:24
• Essentially, the sum of all forces on all particles relative to the center of mass is zero for a rigid non rotating body. This is not true for any point other than the center of mass. – Señor O Feb 26 '17 at 18:26

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.

• This isn't really true. The fact that " transnational kinematics of some body can be determined from considering it as a single body with a total mass equal to the sum of the individual masses" follows directly from Newton's laws. You don't have to say "it's an experimental fact", although that's also true. Really, it's a theorem that follows from the three Laws. – Jahan Claes Feb 27 '17 at 1:37

why mass is additive; i.e. the transnational kinematics of some body can be determined from considering it 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.

The fact that you can consider a collection of particles as one large particle follows from Newton's third law. Taylor gives a nice proof of this in his Classical Mechanics book. I'll redo it here:

Say you have a collection of particles with positions $\{x_i\}$ and masses $\{m_i\}$. They are each being acted on by outside forces, $\{F_i\}$. They also interact among themselves: the force of the jth particle on the ith particle is labelled as $\{F_{ij}\}$. These forces can be thought of as the forces that hold our collection together; each particle might be attracted to every other particle, for example. Then we can consider the acceleration of the center of mass, $X$:

$$\ddot{X}=\frac{1}{M}\sum_i{m_i\ddot{x_i}}$$

This follows directly from the definition of center of mass: $X=\frac 1 M \sum_im_i x_i$, where $M$ is the total mass. Now, we can use Newton's second law to rewrite each $m_i \ddot{x_i}$ as the total force on the ith particle, $F_i+\sum_jF_{ij}$

$$\ddot{X}=\frac{1}{M}\sum_i{F_i+\sum_jF_{ij}}=\frac{1}{M}\sum_i F_i +\frac{1}{M}\sum_{ij}F_{ij}$$

Now, from Newton's third Law, $\sum_{ij} F_{ij}=0$. This is because for every term in the sum, say $F_{ab}$, there is another term $F_{ba}$, and Newton's third law says $F_{ab}=-F_{ba}$. Thus, our formula becomes

$$\ddot{X}=\frac{1}{M}\sum_i F_i$$ or $$M\ddot{X}=F_{total}$$

From this proof, you can see the reason why we can treat extended objects as if they were single particles of larger mass is due to Newton's third law, which allowed us to cancel the sum over $F_{ij}$. Intuitively, this happens because Newton's third law says that no object, not even an extended object, can exert a force on itself. Thus, a collection of particles only accelerates due to external forces.

If you liked this proof, Taylor has a similar one for angular momentum that you might find illuminating.

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!)

This just follows directly from the definition of $I$, as SammyGerbil pointed out.