The Physics Philes, lesson 37: Get Centered
Over the past several weeks we’ve been discussing momentum. This week we’ll continue that discussion, but through a slightly different lens. This week I’ll introduce the concept of center of mass and how it relates to this momentum stuff we’ve been talking about.
The world is made up of weirdly shaped things. Like a human body, for example. Or tree or a chunk or rock or an oxygen atom. An object’s mass is not necessarily equally distributed. Luckily, we don’t really have to worry about all of those complexities when calculating the momentum of a system. We can do this because there is always one point that has the same trajectory as the whole object, as if all the mass of the object were squished down into that one point. And that, my friends, is the center of mass.
As you might expect, the center of mass of an object that has a uniform density is its geometric center. Maybe more surprisingly, the center of mass doesn’t actually have to be within the object itself. Think of a donut. Its center of mass is squarely in the center of the whole.
But how do you find this center of mass? It’s not like it’s going to be labeled for easy recognition. Here’s what you gotta do.
Suppose you have a bunch of particles (as you would when talking about an object on the macro level), and those particles are on a coordinate plane. Each particle has a mass of m1, m2, m3, etc and each have their own (x,y) coordinate. To find the center of mass on the coordinate plane, you need to add the products of the particle mass and the points divided by the sum of the particle’s masses. Do this for both the x and y coordinates, and you’ll get the coordinates for the center of mass. If your familiar with statistics (which I am not), you may recognize center of mass as a mass-weighted average position of particles.
Ok, ok, but who cares? What does this mean? It ends up being a bid deal if the object is moving. It turns out that the x- and y-components of velocity of the center of mass are just the time derivatives of the center of mass’s x- and y-coordinates. The velocity of the center of mass is just the sum of the products of the mass and the velocity’s x-component divided by the total mass of the object. To find the y-component of the velocity of the center of mass, just do the same thing with the y-component of the velocity. If we rearrange the equation, the total mass times the velocity of the center of mass (in vector form) is equal to the sum of the products of mass times velocity, which is in turn equal to the total momentum of the system.
Simply put, the total momentum is equal to the total mass times the velocity of the center of mass. So when you catch a ball, you’re really just catching a bunch of particles. And the impulse you feel when the ball hits your hand? That’s the same as if the ball were one particle that had the same mass rather than a billion little particles.
Crazy, right? But true.
Next week, if I continue to have my things together, we’ll discuss more center of mass motion. Until then, let me know in the comments if I’ve screwed anything up!
Featured image credit: Wikimedia Commons