Dot Physics

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.

![Screenshot 15](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-152.jpg)

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.

![Screenshot 16](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-163.jpg)

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.

![Screenshot 17](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-172.jpg)

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:

![Screenshot 18](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-182.jpg)

Where the gravitational potential energy is:

![Screenshot 19](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-192.jpg)

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:

![Screenshot 20](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-203.jpg)

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:

![Screenshot 21](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-213.jpg)

Now solving for k:

![Screenshot 22](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-224.jpg)

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 (http://www.cabrillo.edu/~dbrown/tracker/) to get positions (I took a screen shot of the game). I get the following:

![Screenshot 23](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-232.jpg)

This gives me a spring constant of:

![Screenshot 24](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-242.jpg)

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:

![Screenshot 25](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-253.jpg)

And, adding the model for a spring:

![Screenshot 26](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-263.jpg)

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)

![Screenshot 27](http://scienceblogs.com/dotphysics/wp-content/uploads/2008/10/screenshot-273.jpg)

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.

Comments

  1. #1 Ambitwistor
    October 28, 2008

    By the way, could you delete the “website” from my above comment (or the comment itself)? Didn’t mean to advertise myself for spam.

  2. #2 rhett
    October 28, 2008

    Ambitwistor,

    I just deleted your previous comment – I didn’t see how to just remove the website. Hope that is ok.

  3. #3 Ambitwistor
    October 28, 2008

    Thanks. Just to repeat my comment, I was suggesting you could write some of your Fantastic Contraption series up for the American Journal of Physics. As Chad Orzel over on Uncertain Principles remarked, it’s an excellent example of the scientific method. It’s interesting because we can’t count on the laws of physics in “simulation world” being exactly the same as ours.

  4. #4 kighillub
    November 26, 2008

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  5. #5 Hanso Kessler
    January 12, 2009

    I find FC a little limited, have you tried Dax Phyz? It’s not really a game, but you can do much more than attach rods to wheels. If FC makes me feel like Newton, Phyz makes me feel like God ;->.

    I think the reason for the perceived springiness of water sticks in FC is due to the iterative nature of the constraint solver, where, in each time-step, the sum of all forces results in movements of connected objects, whose positions then are incrementally adjusted to better fit all constraints (such as stick lengths) in a number of relaxation steps. The number of relaxation steps are most likely fixed to some value like 10 in FC, which means that objects several constraints apart requires more than one time-step to adjust, resulting in soft, springy movement.

    Cheers,
    Hanso

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