How do you find the center of gravity on a given system?
There’s the physical way and there’s the calculus way. There are many ways to describe the physical way but they really end up the same. It’s easiest to describe doing it on something flat… like say, a cutout of the state of Illinois made of construction paper. You stick a single thumbtack in it and let it dangle freely from it. It will hang down eventually into some stable position. Now hang a string from the thumbtack and draw a line down your cutout. Stick the thumbtack somewhere else and repeat the experiment. The two lines will intersect…. at the center of gravity. Ok, the calculus way is to integrate the mass density function of the object times its position vector over the domain of its occupied volume (possibly over all space), and then to divide the resultant vector by the integral of its mass density function over the domain of its occupied volume, NOT times its position vector. That gives you the position vector of its center of mass. For center of gravity, you’d use the
