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How do we get to know the concept of mass in Newtonian Mechanics?

Like, from Newton's Second Law of motion we get : $\frac{d\vec P}{dt} = \vec F$ from here, $m\frac{d\vec v}{dt} = F$, defining $\frac{d\vec v}{dt} = \vec a $, we define mass $m = \frac{\vec F}{\vec a}$.

Once again for a constant $\vec F$, we get, $\frac{m_{1}}{m_{2}} = \frac {v_{1}}{v_{2}}$.

Which one (or any other way of defining mass) is correct physical way to define the concept of mass in Newtonian Mechanics?

User_New2021
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  • Hi, my proposed duplicate marking might be a bit unfair because you might not be asking the same question that is asked in the linked post. But, the top answers there do answer your question, so definitely go through them. Welcome aboard! :) –  May 26 '21 at 19:08

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Place the two masses on a very light glider on an air track, so that they move horizontally with negligible friction. For each mass, pull on the glider with a rubber band, making sure the rubber band is extended to the same length for both masses. Measure the magnitude $a_1$ and $a_2$ of each mass's acceleration. Then the ratio of their masses is \begin{align} \frac{m_1}{m_2} = \frac{a_2}{a_1} \end{align} There is some idealization involved (massless glider, perfectly frictionless and level air track, perfectly elastic rubber band, no uncertainty in the the length of the rubber band or the acceleration), but in principle this is an operational definition of mass.

d_b
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  • Well, I can also define $m_1/m_2 = a_1/a_2$. Why are you particularly defining what you define? More directly, it is unclear what you are assuming and what you are not assuming. If you aren't already assuming Newton's law, I don't find any motivation or justification as to why such a definition of mass is physically interesting or meaningful. If you are already assuming Newton's law then it defies the purpose of what OP is asking which is what is the definition of the mass that enters the Newtonian laws in the first place. –  May 26 '21 at 19:11
  • It is physically interesting and meaningful because this definition corresponds to the $m$ in $F = ma$, but it does not assume Newton's second law. You are of course welcome to define some other property $Z$ such that $Z_1 / Z_2 = a_1/a_2$, or $Z_1/Z_2 = \sin(a_1)/\sin(a_2)$, or whatever else you want, but you won't be able to say $F = Z a$. – d_b May 26 '21 at 19:22
  • Well, sure, but then you are assuming Newton's laws. I don't see how you are not using the second law if you are justifying the physical significance of your definition via $F=ma$. –  May 26 '21 at 19:29
  • This definition manifestly does not assume Newton's second law or any definition of force. (I would need to think a lot harder to decide if it assumes Newton's first law, but I suspect it doesn't.) However, one can show under some mild assumptions that the $m$ output by this definition is the same $m$ appearing in Newton's second law. I don't understand your objection at all. It seems like you are asking "Why define a thing we know is useful, when we could define any number of things that are not useful?" We make definitions because they are useful.... – d_b May 26 '21 at 19:50
  • ... The definition of a group does not refer to or depend on the notion of a group action or of symmetries, but we know that we define a group the way we do because it allows us to talk about symmetry groups. The definition of work does not depend at all on the existence of kinetic energy or the law of conservation of energy, but we know that we define work the way we do because it is useful in connection with those other concepts. From (what I interpret as) your point of view, we should make no definitions which we know ahead of time will be useful, which I find nonsensical. – d_b May 26 '21 at 19:53
  • I guess you are asking "Would we have made this definition if we didn't already know Newton's second law?" Then I guess the answer is no, but I don't see how that is relevant. – d_b May 26 '21 at 19:56
  • Sorry, ignore that now-deleted comment. I didn’t read carefully enough. – G. Smith May 26 '21 at 23:19