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.
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How many floors in that atrium? Is there a skylight?
Spacing between floors for a building like that is typically 10-12 feet in the US, so between 3 and 4 meters per floor. (The distance between the ground floor and the floor above is sometimes a bit higher than spacing between upper floors). Even at the upper limit of 4 m spacing between floors, that pendulum is a bit long for a six-story building. I assume that you follow the US convention of calling the ground floor the first floor: I count four upper-floor railings in the photo, and presumably you were leaning over the sixth floor railing when you took that photo.
Six habitable floors, but there's a large skylight area above those and a further mechanical/AC room above that. The pendulum terminates at a level between those two levels. As such I'd say the pendulum spans ~7-7.5 floor equivalents.
Architectural drawing here, which may clarify.
the tiles appear to be Penrose tiles. five fold symmetry! woo!
http://www.geom.uiuc.edu/apps/quasitiler/
A 500 lb bob in 7.75 g/cm^3 steel is 15.05" diameter. In 19.05 g/cm^3 depleted uranium it is 11.15" diameter. CVD nickel plating is indicated in the latter case.
What directionally unbiased periodic boost makes up for losses?
Nice! Just stumbled across this particular flavor of ScienceBlogs. I especially like the picture from the 6th floor looking down upon the very nicely done Penrose tiling.
This:
http://www.science.tamu.edu/articles/Mitchell+Physics+Buildings+Formall…
Mentions the pendulum as being 102 feet (31.08 meters).
This one says 85' (25.90):
http://www.gostructural.com/magazine-article----the_mitchell_physics_bu…
If we average them together, you're pretty close :-)
Huh, I hadn't seen either of those articles. Thanks for pointing them out! I'm more inclined to trust the latter, as 25.9 is pretty much equal to my 26.0 given the margin of error. It's also the one in the engineering publication, which I'm guessing is more likely to get things right than a university press release.
Uncle Al: I haven't seen it, but I'm told the driving force is an electromagnet. It's directionally unbiased in that it's attached to the freely-rotating pendulum mount rather than the building itself. I can't confirm this though, as the mount is inaccessible as far as I can tell.
Good work everyone on IDing the Penrose tiling! It's unusual in that it's nonperiodic - unlike most other tiling schemes, it doesn't repeat no matter how far out you travel.
Thanks for the explanations. I agree that the 31.1 m figure is implausible for the length of the pendulum, given the descriptions of the building you have provided--I'm guessing that that number is the total height of the building, and the author of the physics department's press release didn't allow for the mechanical room on top or for the finite size of the pendulum bob (which of course has to have some nonzero clearance above the floor).
The central building of the physical science complex at my Ph.D. institution also had a Foucault pendulum, but much less impressive than yours as that building is only four stories high. It, too, had some kind of electromechanical driver, though I don't remember the details.
I'm a bit of a fan of Amman-Beenker tiling. 8-fold symmetry. One day, I will tile a floor somewhere.
a Penrose-tiled floor (or wall) is on my personal wish list for the (unlikely) instance i should ever become wealthy enough to have a house built to my specifications. i'm a bit jealous of that one.
You have a systematic error because you used a value of g that is much too large for your latitude and elevation. I'd suspect that g is less than 9.80 m/s^2 where you are, so 9.80 +/- 0.01 would be better.
BTW, you can get a good estimate of the distance to the ceiling by measuring up from a top floor and timing how long it takes a cell phone to fall to the floor. Estimating the distance up to the "kicker" above the ceiling can be done from the angle the wire makes and the size of the opening.
Have you estimated theta so you can estimate the effect of the anharmonic (theta^3) term on the period? Are you going to blog about the precession rate?
PS - That is one fancy Foucault pendulum, what with the lights showing its precession. It must be interesting to be in a state where they have a dedicated source of funds for Academic Arms Race buildings, funds that can't be used to endow faculty positions. Or is Texas raiding those funds to balance the budget?
Seems like Texas is about to break my irony meter, since they are investing heavily in university-level science while allowing fundies to short-change children's science educations.
CC: Actually, most of the money that built this building was a private contribution from a single very wealthy alumni. He has also endowed quite a few faculty positions including almost our entire astronomy faculty. God bless him. The lights are a nice touch, but they didn't actually come with the pendulum. They were built in-house by the department head (an experimentalist) and his grad students.
The theta^3 correction should be really small, but worth looking at in a future post. More importantly the precession is of course the main point of the pendulum, so I'm saving that for a future post too.
i hope you'll continue making blogs about science. i learned a lot from your blogs.
Was there, at one time in the past, a Foucault Pendulum in the Zachary building on the TAMU campus in College Station?
Steve Althaus, yes, there did used to be a Foucault pendulum in the Zachry Building. I believe however that it was removed sometime prior to 2003 (possibly long before). I'd be interested to find out exactly when and why it was removed if anyone knows. It was located just to the right of the stairs in this image: http://www.beijingren.com/wp-content/uploads/2007/01/tamu1.jpg