Understanding Center of Mass: Properties & Physically Representing

[aaaa202]

I don't feel I have a good understanding of what the center of mass of an object it, and what its properties are. I know it's the position of all mass elements weighted by their mass and divided by the total mass.
I have learned that the center of mass moves as if it was only subject to external forces. So say you pushed on a box of mass m with a certain force F, then the center of mass of that box should move with an acceleration given by F/m. But since the whole box moves with the same acceleration what is then the so special about this specific point?
I also have trouble finding out what exactly the center of mass represents physically. Say it is in the middle of the box, is the center of mass then the point particle situated in the midst of the box or is it a more abstract point?

 

[Oxvillian]

I don't feel I have a good understanding of what the center of mass of an object it, and what its properties are. I know it's the position of all mass elements weighted by their mass and divided by the total mass.
I have learned that the center of mass moves as if it was only subject to external forces. So say you pushed on a box of mass m with a certain force F, then the center of mass of that box should move with an acceleration given by F/m. But since the whole box moves with the same acceleration what is then the so special about this specific point?
I also have trouble finding out what exactly the center of mass represents physically. Say it is in the middle of the box, is the center of mass then the point particle situated in the midst of the box or is it a more abstract point?

The center of mass is just an abstract point in space, representing the "average location" of the mass of the system. There doesn't actually have to be anything there (eg. a donut - the center of mass is in the hole).

Note that all bits of the system don't have to move together. For example, if a firework explodes in mid-trajectory then its center of mass continues to follow the old trajectory (since no external force has acted on it), even though none of the constituent bits do.

 

[Rooted]

The centre of mass is a unique point in a system, almost an average of all the bits of the mass in the system and their distance from the centre of mass. Any forces or torques acting on the system, like gravity, can be thought to act on the centre of mass. For something like a square uniform box, the centre of mass being within the body of the box might seem a bit obvious, and might not seem to make the problem any easier. However if you think about a system like the moon and the Earth for example, the moon is not really rotating about the Earth, they are both rotating around their combined centre of mass. For this system the centre of mass is within the body of the Earth, but in a different system it might be somewhere in the middle of the two. Here's a page about it. http://astro.unl.edu/naap/esp/centerofmass.html

Does that help?

 

[CWatters]

If you accelerate an object it will try to resist as it has inertia. The reaction force is distributed throughout the mass but for most problems it can be considered to act at the center of mass.

If the force you applied to cause the acceleration doesn't act through the centre of mass a couple exists that will try to rotate the object.

 

[CWatters]

But since the whole box moves with the same acceleration...

Not allways true. Ever ridden standing up on a bus? As it accelerates/brakes there is a tendency for you to fall over because the force acting to accelerate you acts on your feet and not through your centre of mas.