Science a GoGo's Home Page Forums General Discussion General Science Discussion Forum The Death of the Centre of Mass Theorem

Many people believe that the Centre of Mass Theorem is a powerful and useful tool in Newtonian Mechanics. In fact it is a farce.

(1) It is trivially true at distances in which the massive object is virtually a point-mass, such as between distant stars.

(2) It is completely incoherent and self-contradictory at distances in which the size of the object(s) approachs 1/20 the distance between them.

Try it yourself. I'll post the simple proof that it is nonsense after a few people try to guess what is wrong.

There is nothing wrong with Center Of Mass Theorem.. it is statistical way of simplifying calculation.. you are free to write all the formulas for all the molecules(if not atom:-))ann then when you will integrate you will find Center of Mass is indeed correct to such an extent that unless you are in a quantum world you will not doubt it.

The exciting thing from a physical point of view is that Newtonian gravitational theory done right also predicts gravitational waves, just like General Relativity.

Why is that exciting rather than irrelevant?

Congratulations, you have discovered that center of gravity is not the same as center of mass!


Sadly, others have already discovered this fact a few centuries ago. So, you won't win the Nobel prize for physics for this discovery.

Let's check that:

Consider a sphere. The center of mass is located at the Geometrical center. It is a fixed point. Let us place a test-particle a few diameters away from a solid sphere on the right along the x-axis. (make the radius of the sphere 1 unit, and put the sphere at the origin). According to the CMT, the mass acts as if it were concentrated at the centre and the distance will be measured centre to centre for purposes of using Newton's Gmm/d^2 formula.

(1) divide the sphere logically into two halves vertically. Each half will have its own centre of mass, located 3/8 radians down the axis of symmetry. The actual position is n't important, but because it is an average mean of the atoms in the solid, it is fixed relative to the geometric skin of the half-sphere.

(2) Naturally, the CM for the other half will be the same distance away from the geometric centre of the sphere in the opposite direction away from the test-particle. The total force will be the vector addition of the two halves. But by inspection this is impossible. The increase in force for one half the mass now located closer cannot balance the decrease for the other half, because while the distances are equal, the forces have changed by an unequal amount. Gravity is an inverse exponential force.

(3) The actual force calculated by summing the halves separately will be larger than the one calculated treating the sphere as a whole. Conversely, the force calculated by dividing the sphere horizontally will be weaker than the 'whole' force, since vertical components from each half will cancel.

(4) The Centre of Mass Theorem contradicts itself, and also the Sphere Theorem as well, which is a special case of the CM.

So as it turns out, the failure of the theorem is not directly connected to absolute sizes, or the discrete localization of mass or charge, but simply the error is directly connected to the ratio of the effective radius of the objects versus the distance between them.

No, I have discovered what I suspected:
That people often confuse and interchange the names of three related but different theorems:
Centre of Mass (CM), Centre of Gravity (CG) and the Sphere Theorem (ST).

We are not discussing the Centre of Gravity concept at all, which only applies in a uniform gravitational field. I could start another thread for that if you wish.