What is the definition of center of mass?

[Bashyboy]

Hello,

I am trying to describe the concept of center of mass, and need some help with particulars:

The center of mass can be thought of the balancing point of a body; it is the point where mass is "evenly distributed" with respect to this point, such that if you were to place the body in a gravitational field and balance on its COM, then the gravitational force acting on each constituent mass would all cancel each and not produce a torque.

The idea I am having most difficulty reconciling is the idea that the mass is "evenly distributed." What is a better description for this, but keeps the same general idea.

 

[phinds]

Center of mass does not (cannot) depend on even mass distribution since if it did ONLY very regular bodies could have a center of mass, which is silly.

I don't know how to describe it formally, but here's how I think of it. If the body in question is assumed to be rigid (and if it isn't, you can get a shifting center of mass), AND you could magically attach something to the center of mass (that magically moves freely OTHER than being attached to the center of mass), then there is no orientation such that the body hanging from the connection to the center of mass would, in a uniform gravitational field, shift from that orientation due to gravity.

 

[Bashyboy]

I guess what I mean by evenly distributed is this: suppose we have a long, thin rod; if you were to divide the rod at its center of mass, each piece of the rod would have the same mass. This idea works for two dimensional objects, I am having difficulty extending it to three dimensional objects.

 

[Doc Al]

I guess what I mean by evenly distributed is this: suppose we have a long, thin rod; if you were to divide the rod at its center of mass, each piece of the rod would have the same mass.

That's true for a uniform rod. What about two unequal balls connected by a thin rod?

 

[Bashyboy]

Hmm, if you were to divide the thing rod at its center of mass, each piece would be unequal in mass.

 

[Doc Al]

Hmm, if you were to divide the thing rod at its center of mass, each piece would be unequal in mass.

Right. So thinking the mass would be "evenly distributed" won't work.

As you already pointed out, you want the distribution such that the net torque about the COM is zero. That's equivalent to saying that the mass-weighted average position of the mass elements would be at the COM.

 

[WannabeNewton]

A hypothetical particle that when the sum of the external forces on the system is zero will constitute an inertial frame (will travel in a straight line with respect to a lab frame or be at rest for all time with respect to a lab frame). This is the intuitive notion of the center of mass that I have personally found the most useful when solving problems in which the center of mass frame becomes invaluable e.g. two body problems interacting under some potential.

Formally it is simply $R = \frac{1}{\int dm}\int rdm$.