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In A.P. French's Newtonian Mechanics (pages $319$, $320$), the author described a way to define a scale for masses. He said,

One can set up an inertial mass scale for a number of objects $1, 2, 3,…$ by finding the velocity changes pairwise in such interaction processes and defining the inertial mass ratio, let us say of objects 1 and 2, by $$\frac{m_{2}}{m_{1}}=\frac{|\Delta \textbf{v}|_{1}}{|\Delta \textbf{v}|_{2}}$$ Similarly for objects 1 and 3, $$\frac{m_{3}}{m_{1}}=\frac{|\Delta \textbf{v}'|_{1}}{|\Delta \textbf{v}'|_{3}}$$ and so on. If $m_{1}$ is the standard kilogram, we then have an operation to determine the inertial masses of any other objects.

The $\Delta \textbf{v}$'s above are the post-collision minus pre-collision velocity, the subscript next to the module indicates the object in question ($1, 2, 3,...$).

He went on,

We must, however, do more. If we let objects $2$ and $3$ interact, then the ratio $$\frac{m_{3}}{m_{2}}=\frac{|\Delta \textbf{v}''|_{2}}{|\Delta \textbf{v}''|_{3}}$$ must be consistent with the same ration obtained from the first two measurements. In fact it is, and this internal experimental consistency then allows us to use the values $m_{2}, m_{3},...$ as measures of the inertial masses of the respective objects.

Question: Had there been an inconsistency with the experiment relating object $2$ and object $3$, i.e. $\frac{m_{3}}{m_{2}}\ne \frac{|\Delta \textbf{v}''|_{2}}{|\Delta \textbf{v}''|_{3}}$, would redefining our scale resolve the inconsistency?

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French is showing that

1) mass is an intrinsic property of just the object, and 2) mass’s effect on kinematic is understood & reliable.

would redefining our scale resolve the inconsistency?

If (1) was true, but (2) was not, then some other way of scaling the mass work work. For example, gravity and a spring scale.

But if (1) was not true, mass was not intrinsic, then there would be no way to assign specific masses to individual objects: that wouldn’t be a meaningful thing.

Violation of (1) is hard to consider, as the idea of mass is so baked into out thoughts. But French was including that as a logical possibility that he needed to rule out.

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