What does it mean that the relationship between material mass and weight is constant and proportional?

Question

I hope someone can help me with this apparently very basic doubt, but I feel like a stupid monkey trying to join two sticks to reach bananas and without success. My English is not good, I am Mexican and I am still learning it.

I am very new to the world of science, and I have started from the basics. With popular books like "Cosmos" and "History of Time", etc. But I really want to get into this and I started with arithmetic. I have started reading a book on arithmetic, and I found this definition:

Weight

It is not possible to determine directly the amount of matter contained in a body; but it is known that the greater its material mass, the greater the attraction that gravity exerts on it, that is, the greater its weight. This relation between material mass and weight is constant and proportional.

Observing the bodies that appear in Nature and mentally separating all their other qualities, to focus only on the attraction that gravity exerts on them, we arrive at the concept of weight. Because of the constant relation that exists between the material mass of a body and its weight, to the point of being expressed with the same number, we will dispense in this work of speaking in a systematic way about the material mass of bodies, to refer only to their weight. But keep in mind that the concepts of material mass and weight are different.

I tried to understand it, and I started to investigate. I skipped ahead several chapters to see the definition of ratio -also relationship-, and I understand that it is a comparison of quantities to know how much one exceeds the other (arithmetic ratio), and how much one contains the other (geometric ratio), and I wondered then how that fits here. Then I had to dig into other books (very advanced in my current state) like an Algebra book where I found the topic Constants and Variables in which there was something called Direct Variation and a definition that goes as follows.

A is said to vary directly to B or A is directly proportional to B when multiplying or dividing one of these two variables by a quantity, the other is multiplied or divided by that same quantity. If $A$ is proportional to $B$, $A$ is equal to $B$ multiplied by a constant. In general, if $A$ is proportional to $B$, the relation between $A$ and $B$ is constant; then, designating this constant by $k$, we have $\frac{A}{B}=k$ and then $A=kB$ this is quite similar to $a=\frac{F}{m}$ and to $w=mg$

I don't want to write so much so as not to bore you. But the point is that I had to jump to something more advanced, because I also had to go dig into Newton's second law, trying to make sense of it. I found something, but I didn't really understand it, I tried to relate it to what I mentioned before and I thought I understood, but when I kept looking I found more about it

One of the important aspects of physics is the search for relationships between different quantities—that is, determining how one quantity affects another.

So if the relationships are comparisons of quantities as I mentioned before, how does it make sense to say that one quantity affects another? Does this have to do with kilogram-force? Is the relationship between weight and mass constant and proportional because it says so $w=mg$? Should I be concerned about understanding this well now, or just settle for an approximation so I don't get confused? Because I just started getting into this.

It's clear that I don't know algebra yet, and also that I'm very confused and stuck. I really hope someone can help me, I will be very grateful.

Answer 1

Not knowing algebra will make studying physics difficult. You can learn a lot of physics with just algebra under your belt (and then a bit more with calculus).

When two quantities are proportional, such as $w$ and $m$ as expressed in $w=mg$, then in a sense they carry the same information and are redundant. You can view two proportional quantities as two expressions of the same quantity, with different units (in this case, $kg$ and $N$, Newtons).

Simple analogy: Let's call the length of something measured in feet $F$, and the length of something measured in inches $I$. These two quantities are proportional: $I=12 F$. I'm sure you can see how they literally express the same measurement in different units. In more abstract examples, the proportional quantities represent different physical properties but can still be viewed as expressing the same information in different units.

Caveat: For the sake of completeness I'll point out that there are a few different degrees to which the constant of proportionality is actually constant. There are strict mathematical constants, such as $12$ in the example above or $\pi$. They are always true. Then there are fundamental physical constants, such as the speed of light $c$ or Newton's constant of universal gravitation, $G$. These appear to be true through the history of this universe as far as we can tell, but we can't rule out the possibly that their values change slowly over time or in extreme situations. The weakest kind of "constant" is one that is only true for special situations. Whenever a "constant" is introduced in an equation, you should try to understand what situations its value will actually be constant for. In the case of $g$, it is only constant near the surface of the earth. At the edge of the atmosphere, or on another planet, $g$ will have a quite different value. It is constant only in special situations (though a lot of useful physics can be done with these special situations, anything that's within a couple miles of earth).

Answer 2

Suppose we hang a bottle of 1 liter of water in one of that spring scales and it displays 1 kg. And it displays 5 kg when a bottle of 5 liters are there.

According to our intuition of mass as related to more or less volume of the same stuff, the scale measures mass.

But instead of using the scale to weight things, it is possible simply to force it using our hands and get a value. So, clearly the numbers can also be a measurement of force.

We can solve the ambivalence by saying that gravity is a force, all the weights are really forces, and proportional to masses. Instead of kg, the proper unit is kgf.