# Simple Spring Systems: Hooke’d on physics

It’s been a brief amount of time since the last post of Joshing with Physics, so to bounce on back let’s talk about some of the simple physics related to a really amazing classical object, the spring!

Hooke’s law is used to determine the force that a spring will exert when pushed or pulled from equilibrium. It is a very simple relationship that surprisingly has no mass dependence!

###### Hooke’s Law

$F = -kx$

Here F is the force, k is the constant of the spring, and x is the distance moved from equilibrium.

Let us be careful to define the two variables mentioned above that are equal to our force. The spring constant ($k$) is how much force the spring will exert for every unit of distance ($x$) we have pulled the spring away from its equilibrium point. The equilibrium point is where the spring would sit while at rest, without any force pushing the spring together or pulling the spring apart.

Let us also consider why Hooke’s law is negative. The force is negative in this case because if we push the spring together the spring will try to return to the equilibrium point and push outward. In this case we have defined the direction inwards of the spring as being the negative direction, so the spring will push in the positive direction and give us a force in the $+x$ direction! Alternatively, if we were to pull the spring apart the spring would prefer to pull itself back together to return to its equilibrium point. In this case we have set the system up to pull with a force in the $-x$ direction!

# But what can Hooke’s force law be used for?

Any force can be defined as a change in momentum, or a preferred change in the movement of an object. Forces can be defined in a lot of different ways such as $F=ma=\frac{dp}{dt}=\frac{dU}{dr}$ and many more! Here m is mass, a is acceleration, p is momentum, t is time, U is potential energy, and r is distance. In our case, Hooke’s law explicitly applies to springs but it is equivalent to all of the other things that a force can be equivalent to. There are so many different things that forces can be related to! Specifically this means that it makes as much sense to say that $ma=F=-kx$! But wait. Didn’t I say that Hooke’s law was not mass dependent? It isn’t! But you can use it to solve for mass if you know the acceleration of a system.

This applies in the case of a spring hanging vertically from a ceiling with a mass on the end. If we know the spring constant of a system ($k$), how much the spring has stretched from its equilibrium point ($x$), and the acceleration (gravity) of our local environment ($a$), then we can solve for the mass! ($m=-\frac{kx}{a}$)

Does it make sense that our mass is negative? I wouldn’t think so! What value do you think is negative so that our mass comes out as a positive value? Let me know in the comments below!

###### Image source: Wikimedia Commons

Hooke’s law is a great and simple way to show a lot of different aspects of physics and gives us some of the rudimentary concepts to begin to talk about things like the bonds between different atoms, as is the case for a 1-D infinite crystalline chain of atoms bound together! Interested in hearing more? Help me continue to write by making a donation below or sending me a happy message! You can contact me by sending an email to (JoshingWithPhysics@gmail.com).

Best wishes,

Josh Lofy