The “Not-so-Center” of Mass

This post is by Student C, Math 493.

What is the “Center of Mass”? Consider a system of n objects in 1 dimension. Then the center of mass follows this set of relations. Let Xi be the position of the ith object in our one dimenion. Then the center of mass is located:

X = sum(m_i,x_i) / sum(m_i)

But what does this concept of center of mass actually represent?

The center of mass of an object should be obvious. Let's redefine the center of mass. The center of mass is a place where, if you put your finger on that spot, you should be able to balance the object on your finger. Similar to the bird below:

Notice how this is at a state of equilibrium when placing your finger at the center of mass.

So what is not the center of mass? Is that where I can place my finger on the object and have the object fall off my finger? That is, if I were to place my finger on the wing of the bird, in theory, it should fall off my finger. Many readers, one would hope, may have had experience with these birds. To which you would say that the bird would simply fall to the ground if you tried to balance it on anywhere but it's beak. The reason for this is simply because you are able to, experimentally, calculate the center of mass of the bird and place your finger in such a way that the bird balances on your finger.

With this in mind, let us try to expand this notion of where we can place our finger. What if I placed my finger completely off the bird. Could it balance? Could such an object even exist? One where it's center of mass is not on the object. Intuitavely, this makes no sense as there is no purchase for my finger to go in order for the center of mass to exist. In comes a very simple object:

Note here, that neither one of the objects are being placed at their center of masses. Or, these two objects are balancing on their not-so-center of masses. So it seems that the solution here was to add another object.

This then begs the question. Can more than two objects have this relationship? What about an arbitrarily large amount of objects? What about an infinite number of objects? The best answer to this question is by challenging infinity itself and asking another very deep-seeded question- is there a center of mass for the universe? And if so, where is it? Consider this “proof” by contradiction

Suppose for a moment that there was a center of the universe. Besides the fact that we would have to re-write all physics text books, what would happen if I placed my finger there? Am I balancing the universe? Can I move the universe? And if I don't place my finger there, is the universe in a state of unbalance? In this manner, we have reached a contradiction.

One might conjecture that the center of the universe would be at the “location” where the big bang originated. This is a closer approximation. The fallacy here, however, is assuming that the big bang occurred at a location. To think about how this is done. Consider trying to find, in 3-dimensions, a location for this big bang. Under the assumption that all objects are radially moving away from a central point, we can follow these paths back to the “origin”. However, the most surprising thing happens when you do this. Wherever you started, when doing this process, is where this process will end up. Coming to the conclusion that the center of mass of the universe is everywhere at once.

Citations:
Center of Mass equation - https://en.wikipedia.org/wiki/Center_of_mass

Bird photo - https://en.wikipedia.org/wiki/Center_of_mass

Motivation - http://map.gsfc.nasa.gov/universe/uni_shape.html

Appendix
For the wine bottle holster. The equations that were used was assuming that we had a uniform rectangular prism. That is:

xcm = sum(mx) / M

ycm = sum(my) / M

zcm = sum(mz) / M

Next, using the same equations, we removed the triangular prism and cylinder from the rectangular prism. This generates the center of mass of just the holster. Experimentally, one can find the center of mass of the wine bottle and adjust how much/little you remove from the rectangular prism to make the center of mass an equilibrium point.