Water Stick springs in Fantastic Contraption (the game)

Here is what is cool about [Fantastic Contraption](http://fantasticcontraption.com/) - it's like a whole new world, a world ready for exploring. I am Newton, and I can see if this world follows the models that I propose.

In this post, I am going to explore the elastic nature of the "water-sticks". If you have played fantastic contraption, I am sure you noticed that the water-sticks are springy. How does these springy sticks work? Are they just like the springs we have in the real world? An excellent model for springs in the real world is Hooke's law. It says the force exerted by a spring is proportional to its stretch.


Obviously, this is the magnitude (not the actual force, because that would be a vector). k is the "spring constant" or the stiffness of the spring (in N/m). s is the amount the spring is either compressed or stretched from its natural length. The minus sign is sort of silly. It is there to show that the force exerted by the spring is in opposite direction as the stretch.

Another important aspects of springs (in the real world) is the energy stored in a spring.


So, now what about the FC-world (Fantastic Contraption)?

To explore this question, I created a machine that has a ball falling while attached to a series water-sticks.


I will analyze this in terms of energy. As the ball drops, the system consisting of the ball, the water-sticks and the Earth (or whatever planet it is on) will have constant energy. There is no external work on the system so:


Where the gravitational potential energy is:


It doesn't matter where *y* is measured from since the only thing that shows up is the CHANGE in potential. So, what two positions will I consider? I will consider position 1 to be right when the ball is released. Position 2 will be when the ball reaches its lowest point. These are nice points to choose since the kinetic energy for both cases is zero. This gives an energy equation of:


s₁ is zero (it starts off with no stretch). I also will place the origin at the lowest point such that y₂ is also zero. This gives:


Now solving for k:


I can get values for everything except the mass of the ball (well, I can get the mass in terms of mass of the ball - like I did before). I will use [video tracker](http://www.cabrillo.edu/~dbrown/tracker/) to get positions (I took a screen shot of the game). I get the following:


This gives me a spring constant of:


Ok, but I really didn't test if the water-sticks obey hooke's law (since I only have one data point). I could repeat the experiment, but drop it from a different height and see if I get the same spring constant. (I will leave that as an exercise for a student) There is one other way I can test this spring with the set up I have. After the mass stops bouncing, it is equilibrium. The final stretched length of the water sticks is 4.61 U. If Hookes law is working here, then the upward force from the spring should be the same as the downward force of gravity:


And, adding the model for a spring:


Ok - not the same thing. Something weird is going on. Truthfully, I already knew this. Suppose I replace the many small water sticks with two larger ones (of about the same total length)


It essentially does not bounce at all. I have an idea that the water sticks ARE NOT springy. Perhaps it is the joints between sticks that are springy. This would mean that this last set up has very few springs where as the previous had a lot.