Happy Pi Day.
I was trying to think of something cool to do for Pi Day. Last year, I made a numerical simulation to determine the value of Pi. This year, I was stumped. What could I do that was different? My first idea was ok, but not that great. The idea was to measure the circumference and diameter of a whole bunch of objects. Then plot circumference and diameter with a linear fit. The slope of this line should be Pi. I know, this is an important activity - but not very spicy.
When people think of pi (or should I write π?) they either think of pie or they think of circles. So, let me determine the value of pi without using circles and without using a pie. Really, this is what is so cool about pi. It is more than the ratio of circumference to diameter for circular objects. That is just scratching the surface.
Measuring Pi with a Spring
An oscillating mass on a spring. It seems so simple, but it is so important in so many ways. There are two important things about springs. First, when you pull or push on one, the force it exerts is proportional to the amount of stretch (or compression). Typically, this is called Hooke's Law and can be written as:
Just to be clear, this is the magnitude of the force the spring exerts (so I left off the negative sign). The k is the stiffness of the spring (spring constant) and s is the amount the spring is either stretched or compressed.
Suppose I take the same spring, but a mass on the end and attach it to a wall. I then pull the mass back on a smooth frictionless surface and let go (you could do this with a mass hanging on a spring too). The motion of the mass can be modeled with the following:
This in of itself is pretty cool. I mean, look at the mass-spring system. Do you see a right triangle? Do you see a circle? No. But there is a trig function right in there. Oh, I should point out that in the above expression, A is the amplitude of the oscillation and ω is the angular frequency. In order for things to work out, the following must be true:
T is the period of motion, the time it takes the mass to make one complete oscillation. How do you get all this stuff from the motion of a mass on a spring? Here are is a post on a solution to the mass on a spring.
Measuring the spring constant.
Maybe you can see where this is going. I will first determine the spring constant k. After that, I can measure T for different masses. From that I can solve for π.
Here is a mass hanger on a vertical spring. As I add masses, I record the position of the bottom of the mass holder (not including the mass of the holder). The one nice thing about Hooke's law is that I can write:
If I plot the force the spring exerts (which would be the weight of the stuff hanging on the end) vs. the position, the slope will be the spring constant. The y-intercept doesn't matter. Here is the data:
From this, I get a slope - and thus a spring constant of about 3.04 N/m. I could probably get a more precise value, but I did this rather quickly.
Period of Oscillation
Perhaps for the next step, I could just put a mass on there and find the period of oscillation. No. Not good enough. Instead I will do several masses and make a plot of T vs. the square root of the mass. According to the expression for the period above, this should be linear. Let me write this as:
From the slope of this plot, I can get π. Here is the data:
Yes. I know there are only 4 data points. I was a bit rushed when I collected the data. Also, sorry for using Google docs. It is nice and quick, but it does not create linear fits all too easily (but it can be done). However, if you find the slope of this line you get:
So, 3.29 is not a very good value for π. However, the point is that it could be done. You can determine π without using a circle or even a right triangle. Sure, I could have done a better job. This was just a proof of concept.
