Built on Facts

Why the long discussion about the period of a pendulum yesterday? Because we're actually going to take a look at a particular pendulum today. This one hangs in the central atrium of the George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, which constitutes half of the beautiful and brand spanking new two-building complex now housing the Texas A&M; department of physics.

The pendulum is a Foucault pendulum, meaning its support allows it to swing freely in any direction. If you put one of them at the north pole, the plane in which is swings would appear to gradually rotate, making one complete rotation in one day. In reality it doesn't, it just appears to as the earth rotates under it. Conversely, at the equator a Foucault pendulum stays put because the earth isn't rotating with respect to the plane of the pendulum's motion. In between the poles and the equator, the rotational period varies with latitude. At the latitude of College Station, Texas, the period conveniently happens to be just a hair under 48 hours.

But that's a story for another time. We're not interested in the rotational period, we're interested in the actual period of the back-and-forth swing. The equation we derived yesterday tells how the period of the swing is related to the length:

It's easy to measure the period of this pendulum, all you need is a stopwatch. But what if we were curious about the length of the string? That's pretty hard to measure, given the length and inaccessibility of the string. Here's the view down from the 6th floor, the pendulum actually continues upward several more meters before disappearing into the ceiling:

[Bonus Question: What's the name and mathematical significance of the tile pattern?]

But we can calculate the length by inverting the above equation:

I measured the pendulum time with the stopwatch on my cell phone, timing 5 complete cycles and then dividing by 5. I did this 6 times to get an idea of the variance. The results for the period of the pendulum were {10.254, 10.212, 10.226, 10.226, 10.212, 10.240} seconds. The average is 10.228s, and the standard deviation is 0.016s. I expect the repeated values represent hidden discreteness in the cell phone timer, but you work with what you've got.

Plug in the average for T and the standard value of 9.81 m/s for g, and we get:

L = 25.9951 m

This is absurdly precise because we've just plugged in without accounting for possible error. We should do this. The error in a measurement due to the uncertain quantity is generally taken to be the rate at which the dependent variable (L here) changes with respect to the unknown quantity, times the uncertainty in that quantity. If there's more than one observed quantity with uncertainty, you repeat this for each variable and add using the rules for error propagation. (Here I assume the uncertainty in g is 0.01m/s^2, and use g = 9.81 m/s^2.)

Plugging all that in, the total uncertainty in L is 0.0855378s. Rounding off, the total length of our pendulum is:

L = 26.00m ± 0.09m

Now we have to worry a little. There's two kinds of error. There's statistical error, which is a result of the intrinsic fuzziness of having fluctuations in our imperfect measurements. We've dealt with that here, and we know roughly how much uncertainty in the length we get from that statistical noise - about 9 centimeters. But there's can also be systematic error, which is a result of an overall bias in the measurement. I don't mean bias in the sense of skewing the results to our own prejudice, I mean bias in the sense of there being something wrong other than just the statistial noise. For instance, imagine you're driving down the interstate at 70mph on your speedometer. The terrain and your foot reflexes prevent you from maintaining exactly 70 on the speedometer - that's statistical error, in a sense - but if the speedometer itself is off by 10 mph then that systematic error means your precision is for naught. And you might not ever even find out. Systematic error is insidious and every experimentalist spends lots of time thinking about how to track it down and make sure it's all accounted for.

What might be the systematic error here? Well, for one thing our pendulum has a finite amplitude and so there's error due to the sin(x) = x approximation we used to derive the equation for the period. The amplitude is very small however, so this correction will be too. If we were feeling especially brave we could eyeball the horizontal swing and estimate the angle and do the correction ourselves without a ruler, which would at least give us an idea. Then there's the fact that the pendulum string is not massless but is in fact a steel cable. It may or may not be a significant fraction of the mass of the bob at the end, resulting in a non-ideal pendulum (or maybe not, as it turns out the bob is solid metal and is nearly 500 pounds). Then there's the fact that the string itself vibrates very slightly but visibly with the air currents in the room. Then there's the fact that the bob isn't a point mass in the first place, meaning we have to be very careful with the close-but-no-cigar assumption that the effective L is in fact the length of the strong. And so on and so forth.

If we were writing this up for formal publication, we'd have to sit down and work out all those things and anything else we could think of. As it is though, I think we're in pretty good shape. The pendulum is 26 meters long, give or take some quantity that's probably a small fraction of the total.