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If I look at Hooke's law as it's defined in my textbook, it looks like:

$F = -k\Delta s$

Therefore, the restoring force of an ideal spring will be proportional to the displacement from equilibrium, where $k$ will be the constant of proportionality. From this equation, I believe that it can't be said that mass is proportional to the displacement from equilibrium (even though it seems to be the case implicitly).

However, if I substitute $ma$ for the restoring force:

$ma = -k\Delta s$

Is it then valid to say that mass is proportional to the displacement from equilibrium, as well as mass is inversely proportional to acceleration? I've been thinking about proportionality a lot lately and this one sort of threw me for a loop.

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In a sense yes, if you're very careful about what you're holding constant. Stating that a variable is proportional to another variable implies that all other relevant quantities are being held constant.

For example, there's a simple relation $d=vt$ that describes the distance $d$ something travels in a time $t$ when traveling at speed $v$. One might say that $d$ is proportional to $t$. However, this relation is only valid if the speed $v$ is constant throughout the interval $t.$

In your example, one could say that mass is proportional to displacement if the $k$ and $a$ are constant, but you will need to do some work to figure out what physical system(s) meet such requirements.

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  • $\begingroup$ All of my confusion disappeared after reading this post. Thanks a lot! $\endgroup$ – user32297 Nov 13 '13 at 5:16
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You have to be a little careful to make sure you're clear on what you're modeling when you combine various equations together.

If you are thinking of a mass attached to a spring attached to the ceiling, then yes, the mass is proportional to the spring displacement. This follows since in the case of an object being acted on by gravity, $a=g$, and thus $|\Delta s|=mg/k$. This should make intuitive sense; if you hang a heavier weight, it should stretch more and hang lower.

In general, however, the exact form of $a$ is not specified in the question, so to make any sound statements you'll need more info on what the system you have in mind is.

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