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.

 

[PJC01]

I can confirm that there is indeed a difference between the center of mass and the center of gravity. The center of mass refers to the point at which an object's mass is evenly distributed, while the center of gravity refers to the point at which an object's weight is evenly distributed. In most cases, these two points are the same, but there are certain situations where they can differ.

One example is when an object is in a gravitational field that is not uniform, such as on a tilted surface or near a massive object. In these cases, the center of mass may not align with the center of gravity. This difference can have practical implications, such as in the design of spacecrafts or in understanding the stability of structures.

As a teenager, it may not be necessary to fully understand the differences between the center of mass and center of gravity. However, it is important to be aware that they are not always the same and can have significant effects in certain situations. As you continue to study science, you may encounter more complex concepts and applications where understanding these differences will be important.