# Simple Spring Systems: Hooke’d on Negative Mass

Have you already read the recent Joshing With Physics post “Hooke’d On Physics“? I left an open ended question that asked how could the mass in Hooke’s law be negative, as shown here, $m=\frac{-kx}{a}$? $k$ is the spring constant, $x$ is the distance traveled, and $a$ is the acceleration experienced (in this case, gravity).

This question arose from the presentation of Hooke’s law as $F=-kx$. Hooke’s law was represented this way because if you were to pull the spring in any direction $x$ it would pull in a negative direction $x$.

We set Hooke’s law equal to Newton’s force of $F=ma$. This representation can be shown as $ma=-kx$ and simplified to the previous expression $m=-\frac{kx}{a}$. However, the mass cannot be negative! So what is our mysterious negative entity here?

###### Image Source: Wikimedia Commons

If you guessed gravity, then you just don’t understand the gravity of the situation. In this representation the gravity $a$ and $x$ are in the same direction. Even though in many physics representations we represent gravity as a negative value, we do so because we have defined the ground to be zero and upwards (towards space) to be the positive direction.

So our mysterious force is actually Hooke’s law itself! Often referred to as the restoring force. In the case of solving for the mass of an object we are not concerned with the restoring force of the spring, but the force that the mass has on the spring itself.

Considering the setup of any physics problem, specifically why is something negative or positive in your free body diagram, is always more important than just plugging and playing with the memorized equations in any study guide. It is always important to consider the entirety of the problem so you can arrive at a much more sensical answer.

Thank you for your continued support in reading my blog! More follow ups are coming from previous topics. As well, I recently was able to acquire a Solid State Physics text. My future blog posts will be featuring the fascinating physics of crystals! From metals, non-metals, and beyond.

Best wishes,
Josh Lofy