I understand the concept of Center Of Mass(com), but I am having a difficult time interpreting the equation of the simplified case of one-dimension.
The book I am reading defines the position of the com of a two-particle system to be $x_{com}= \Large\frac{m_1x_1+m_2x_2}{m_1+m_2}$ I'm sorry if this seems like a trivial question, but could someone explain to me the interpretation of this definition? Perhaps even why they defined it to be this way.
Here is an excerpt from my textbook:
"An ordinary object, such as a baseball bat, contains so many particles (atoms)that we can best treat it as a continuous distribution of matter. The “particles” then become differential mass elements $dm$, the sums of Eq. 9-5 become integrals, and the coordinates of the center of mass are defined as (9-9)...Evaluating these integrals for most common objects (such as a television set or a moose) would be difficult, so here we consider only uniform objects. Such objects have uniform density,or mass per unit volume; that is, the density $\rho$ (Greek letter rho) is the same for any given element of an object as for the whole object. From Eq. 1-8, we can write $\rho = \large\frac {dm}{dV} = \frac {m}{V}$"
What does the author mean by, "the 'particles' then become differential mass elements $dm$? Is $\rho = \Large\frac{dm}{dV}$ the derivative of the density function? If so, how would I interpret that? Furthermore, if it is indeed the derivative of the density function, why is it also equal to the original function of density?