What is the center of mass?
The center of mass is a position defined relative to an object or system of objects. It is the average position of all the parts of the system, weighted according to their masses.
For simple rigid objects with uniform density, the center of mass is located at the centroid.
For example, the center of mass of a uniform disc shape would be at its center. Sometimes the center of mass doesn’t fall anywhere on the object. The center of mass of a ring for example is located at its center, where there isn’t any material.
What is useful about the center of mass?
The interesting thing about the center of mass of an object or system is that it is the point where any uniform force on the object acts. This is useful because it makes it easy to solve mechanics problems where we have to describe the motion of oddly-shaped objects and complicated systems.
For the purposes of calculation, we can treat an oddly-shaped object as if all its mass is concentrated in a tiny object located at the center of mass. We sometimes call this imaginary object a point mass.
If we push on a rigid object at its center of mass, then the object will always move as if it is a point mass. It will not rotate about any axis, regardless of its actual shape. If the object is subjected to an unbalanced force at some other point, then it will begin rotating about the center of mass.
How can we find the center of mass of any object or system?
In general the center of mass can be found by vector addition of the weighted position vectors which point to the center of mass of each object in a system. One quick technique which lets us avoid the use of vector arithmetic is finding the center of mass separately for components along each axis. I.e:
For object positions along the x axis:
Together, these give the full coordinates (\mathrm{COM}_x, \mathrm{COM}_y)(COMx,COMy) of the center of mass of the system. For example, consider the system of three flat objects of uniform density shown in Figure 2.
The location of the center of mass in the xxx direction is:
