Unit analysis of classical equations, and why charge and mass may be manifestations of the same thing?
Consider <PERSON> force for universal gravitation, $$F_g=G\frac{m_1 m_2}{r^2}.$$ Nothing new here, but bare with me. This is a strange idea, hard to articulate adequately without a greater understanding of physics which I do not afford, but is there any merit to conclusions we can draw from the unit analysis of this equation when letting $G=1$?
That is to say, suppose $G$ is unitless; just a conversion factor consequential of our unit system and that such an equation really does tell us about some subtle nature of units. To see what I mean, follow me with this example:
Converting this into an expression of units, we find
$$|M||L||T|^{-2}=|M|^2|L|^{-2}\implies |L|^3=|M||T|^2,$$
Now, to me, length and time are quite meaningful units to me. I could not fathom how either could be dependent on some other (more fundamental) units. 298
Mass however, has always been a unit which puzzles me, and feels like it is constructed from other more fundamental units.
With the above equation, (under our controversial assumption), we find $$|M|=\frac{|L|^3}{|T|^2},$$ that is, $\text{Mass} = \text{Volume per Time per Time}$. Strangely, this seems rather meaningful. I could certainly imagine mass as a resultant property of something to do with volume and a rate of change of said volume.
Does this unit analysis have any merit or deeper meaning, or am I just mad?
Furthermore, it would seem that each (albeit classical) equation for the fundamental forces, yields additional definitions of units, under similar assumptions. For example, <PERSON>'s Law $$F_e=k_e\frac{q_1q_2}{r^2}\text{, yields }|M||L||T|^{-2}=|Q|^2|L|^{-2}\implies |M||L|^3=|Q|^2|T|^2,$$ when $k_e=1$.
Much like mass, charge too puzzles me as seemingly a non-fundamental unit. Thus we could deduce that $$|Q|^2=\frac{|M||L|^3}{|T|^2},$$ which when combined with our previous equation for mass, yields $$|Q|^2=\frac{|L|^6}{|T|^4}\implies |Q|=\frac{|L|^3}{|T|^2}.$$ Hence, $\text{Charge} = \text{Volume per Time per Time}$.
Could this be a hint that charge and mass are manifestations of the same thing, exhibited in different ways? 453