The answer is yes. Yes, you can get a value of pi using a pendulum. Well, you need a few other things. Of course, this is the experimental part of my previous post on the connection between pi and the gravitational field, g. In that post, I basically said that the period of a pendulum (with a small amplitude) and length L is:
Further, I said that the period of a pendulum with a length of 1 meter is 2 seconds. This would mean that pi squared would be g (the gravitational field in N/kg) - which it is.
Oh, well that is just a coincidence. NO! It's not. It's not magic either. Well, it's not magic but I think it's magical.
Here is the plan. I am going to measure the gravitational field (g) using some arbitrary distance units (not meters). Next I will measure the period of a pendulum and record the length in these same non-meter distance units. From those two experiments, I will calculate pi. Can't be done? I actually don't know for sure this will work, so just be patient.
Measuring the Gravitational Field
Like I always say, it's not the acceleration due to gravity. It is much more appropriate to call it the gravitational field. However, for a free falling object (one with only the gravitational force on it), the vertical acceleration has the same magnitude as the gravitational field. But no, it's still not the acceleration due to gravity. Ok, maybe just this once you can call it that - but don't do it again.
Here is a high speed video showing a ball shot up vertically (240 frames per second).
Yes, there is a meter stick taped to the wall - but I'm not going to use it. Instead, I will measure distance in units of "blocks". One block is the height of one of the cinder blocks in the wall. Hopefully, this distance is standard enough for my calculation to work.
Using Tracker Video analysis, I can get the x and y position of the ball after it is in the air. I wasn't completely sure about the vertical direction in the video, so I am going to calculate the acceleration in both directions. Here is a plot of the horizontal position.
I can compare this to the following kinematic equation:
This means that the fitting coefficient in front of the t2 term is going to (1/2)a. This ball has an x-acceleration of -0.042 blocks/s2 (blocks instead of meters).
Here is the plot in the y-direction.
This says that the y-acceleration is -51.22 blocks/s2. Ok, let me assume that the actual horizontal acceleration is zero. That means that vertical acceleration will be the total acceleration (remember, x is not exactly horizontal). I can find the magnitude of the acceleration from the components of acceleration.
The total acceleration is then 5.7462 b/s2 (the b stands for "blocks"). This is very very close to my y-acceleration so I my guess for the direction of vertical was not so bad.
Then what is the value of the gravitational field? Let's call it g = 51.22 Nb/kgb. See what I did there? I made a new unit. The gravitational field is in units of block-Newtons (Nb) per block-kg (kgb). This has the equivalent units to b/s2. Also, I probably should do this experiment several times and get an average - but I won't. You can do that for homework. I am just trying to get a proof of concept.
Period of a Pendulum
I'm not going to use a seconds pendulum. Well, I could but it wouldn't be 1 block long. Instead let me look at the the relationship between period and length. I think I could get better data than what I have, but it wouldn't be as quick. Here is a video where I have a swinging pendulum. As the pendulum oscillates, I change the length. All from this video I can get several values for length and period.
If I load this sucker into Tracker, I can get the length and the period. Here is the data I get. Oh, I am using units of blocks again. I will assume that the blocks in this room are the same as in the hallway from the other video.
Here is the data as a Google Docs Spreadsheet in case you are curious. Remember, the length of the pendulum isn't in meters but in blocks.
I always like to make linear plots. If I plot the period squared vs the length, I can write the period equation as:
From this, I can see that the slope of this line should be:
Since I already have an expression for g, I can get the slope and solve for π. Let's just check something. What about units? The slope of T2 vs. L should have units of sec2/blocks. If I use units of blocks/s2 for g, then we can see that the units for this slope should work out to what I expect.
Now for the plot. Here is T2 vs. L.
The slope of the linear fit is 0.8288 s2/blocks. Now for the calculation of π. In case it isn't clear, I am using m to represent the slope.
There you go. π = 3.257. Yes, this is a little different from the accepted value - but I think my method works. I didn't use a circle and I didn't use a meter. I still got something close to π. I could make this better though. First, I think I could do the projectile motion thing many more times and get an average for the vertical acceleration. Second, I need a better unit for length. The block unit maybe isn't too reliable. What I should have done was to just get some stick and declare that as my length unit. Oh, the pendulum data could have been better too. On several of those swings, I only had a couple of (or one) oscillation to get the period.
Home Page Photo: CCAC North Library / Flickr
