Centre of mass and centre of gravity

[asdff529]

Hi,today my teacher said that there is a little bit different between centre of mass and centre of gravity.
But he said there is no different between them in my level(teenager)
What is the different between them and why we can ignore the differences?

 

[D H]

The center of gravity (not necessarily unique) of some object is a point such that the gravitational force acting on a point mass with the same mass as the object and located at that point is the same as the gravitational force acting on the object.

The center of mass of some object has nothing to do with gravity. It is a weighted average of the position of the object, where the weighting is by mass in the case of a collection of objects, or by density in the case of a continuum.


To illustrate the difference, consider a long rod of mass m and length L that is oriented vertically such that the bottom of the rod is a distance r from the center of the Earth. The hypothetical space elevator is a good example. The center of the mass is located halfway up the rod at a distance r_CoM=r+L/2 from the center of the Earth. With a little but of calculus, the gravitational force acting on the rod is GM_Em/(r(r+L)). Thus the center of gravity is given by r_CoG²=r(r+L)=r²+rL. Note that this is below the center of mass. If the rod is very long (e.g., a space elevator), the center of gravity will be well below the center of mass.

 

[HallsofIvy]

Thanks, D.H. If I understand what you have said correctly, if the gravitational force were constant, rather than depending on "$1/r^2$", then "center of mass" and "center of gravity" would be exactly the same. Of course, if, as in most problems, the distances are short enought that gravitational force is constant to a good approximation, then center of mass and center of gravity are the same to a good approximation.

 

[D H]

The center of gravity per the definition I gave (not mine; it is a fairly standard definition) is not well-defined in a uniform gravity field. The gravitational force is tautologically the same everywhere in a uniform field. This means any point will do, but it is customary to pick the center of mass as the center of gravity.