The Test(ing) of Time: The Surprisingly Good Hourglass

My class this term is a “Scholars Research Seminar” with the title “A Brief History of Timekeeping,” looking at the science and technology of timekeeping from prehistory through modern atomic clocks. This is nominally an introduction to “research methods,” though the class operates under a lot of constraints that fully justify the scare quotes, at least for scientists. As I am a scientist, though, I want the class to include at least one original measurement and the reporting thereof, so I’ve been thinking of really simple measurements that I can have them do independently and write up. I’ve previously blogged about some measurements using cheap timers and the NIST web site, that I did as a preliminary for this class.

In poking around a bit, I stumbled across this European Journal of Physics paper on the physics of a sand glass (thanks to a passing mention of it in The Physics Book), which is a really nice piece that we’ll be going over in detail in class. This suggested another possible cheap and easy measurement, though, so I bought a pack of cheap sand timers from Amazon, and intend to hand them out (along with digital timers from our teaching labs and the NIST time URL) as another possible measurement.

i-bfcad2807a36586071214f6f2a193fb5-hourglass.jpgBecause it would be the height of foolishness to assign students to make a measurement that I haven’t done myself, I pulled one out of the pack (shown at right), and tested it myself. The testing protocol was extremely simple: I fired up the stopwatch app on my Android phone, and measured the time required for the timer to empty, flipping it back and forth until I had 32 total measurements (these were spread over the course of a couple of days, in intervals when SteelyKid was watching tv or otherwise distracted). Averaging together all 32 measurements, I come up with an average emptying time of 177.5+/-0.6 s, where the uncertainty is the standard deviation of the mean of the 32 measurements (which were rounded to the nearest second).

That by itself is better than I expect for cheap plastic timers that sell for less than $1 each– the uncertainty in the time is about 0.3% of the time, which is pretty darn good. But it’s actually much more interesting than that, if you dig into the data a little. Here’s a histogram of the emptying time for the various measurements, with the height of the bars indicating the number of trials falling in each 1-s bin:


Pretty striking, isn’t it? This is a classic example of a bimodal distribution– the average of all the trials is 177s, but there were basically no runs that gave a time of 177s. Instead, there was a big clump at around 174s, and another big clump at 180s.

What’s going on here? Well, as you can tell from the bar coloring, the two different groups correspond to the two different orientations of the timer. One end has a “made in China” sticker on it, and when that end is up, the emptying time was 174.7+/-0.4 s (mean and standard deviation of the mean for 16 trials). When the sticker was down, the emptying time was 180.3+/-0.3 s (mean and standard deviation of the mean for 16 trials). That’s a difference of 14 standard deviations, which even a particle physicist would accept as a significant separation.

So, what accounts for the difference between the two? I think it’s the cheapness of the timers. Specifically, the plastic end caps on the tube containing the glass bulbs– one of them is not on quite straight (the one without the sticker), and as a result, the tube is slightly tilted in one orientation relative to the other. Weirdly, the orientation that, by eye, appears to be tilted from the vertical is the one that empties faster; I would’ve expected the more vertical of the two to be faster, but maybe the dynamics of the sand flow process are a little counterintuitive in this respect. (For the record, tilting it more dramatically (by propping it on the handle of a spoon) did significantly slow the emptying time, as expected.) Or maybe it’s just really difficult to judge the verticality of the timer.

This does, of course, suggest another possible experiment, which wasn’t included in the otherwise fairly exhaustive EJP paper, namely measuring the effect of the tilt on the emptying time. I’ll leave that for an interested student to try out, though, and just note that when you start looking closely at the physics of things, even cheap plastic made-in-China crap holds some surprises…

10 thoughts on “The Test(ing) of Time: The Surprisingly Good Hourglass

  1. The same thing happens when you empty a volumetric flask- full vertical is substantially slower (and messier) than a slanted pour. The way I always thought about relative vertical orientation vs. speed of emptying was that you need counter-currents to empty the flask- air runs into the flask as liquid runs out. If the forces on the fluid from all sides of the aperture are equal, then inflowing current is necessarily disruptive. If the forces on one side are greater than the other, laminar outflow can be established on that side, and inflow can occur on the other. Of course, applying this to the hourglass requires that the sand behave like a liquid, which to my understanding is only sometimes true.

  2. Don’t you think the sticker up/sticker down difference could rather be due to a sligth asymetry in the shape of the hourglass ? A tiny asperity on one side close to the central part (but not in the central tube) could slow the flow down in one direction without impacting the other direction I think.

  3. Did you check the level of the surface on which you were testing? Even if it wasn’t level compared to the testing surface (a table or desk?) might not be level, and the “tilted” side could more in line to gravity, and the “level” way could be tilted. Or both could be off.

  4. Don’t you think the sticker up/sticker down difference could rather be due to a sligth asymetry in the shape of the hourglass ?

    It might be. I guessed that it’s the tilt because it does appear to be slightly tilted in one orientation compared to the other. It could be a slight asymmetry, though, or a combination of both– running slightly slow because of the tilt, but somewhat faster because of the asymmetry. Figuring that out would take a lot more effort than I’m willing to put in right now, though.

  5. Why are you using the standard deviation of the mean as an estimate for the experimental uncertainty when you took only 1 sample of 32 measurements? The sample standard deviation seems like a better measure, and would give you a peak separation of about 4 standard deviations for your bimodal distribution, which looks more consistent with your plotted histograms. Similarly, the timing error for a single run is then just over 2%, which seems more realistic.

  6. As to the different “draining times.” My thought is that if the tilt from vertical is slight the sample times when the hour glass is tilted should be measurably less because there is slightly less congestion at the aperture. Perfectly vertical means all grains of sand have an equal chance of winning the race to the drop hence there is more bumping and jostling which delays the winners making it to the drop.

  7. Whether it’s different tilt or different shapes of the approach to the bottleneck, I expect the underlying physics to be the same. As sand flows out of the top, the space it vacates must ultimately be replaced by air from the bottom. So while the sand is falling, there is an air pressure differential pushing back against gravity. If the air from the bottom has a clear path to the top, the pressure equilibrates more quickly and the sand falls faster. But if the only air path to the top requires capillary flow through the sand, the air pressure differential will be maintained until the sand has fallen completely to the bottom.

    I have assumed above that the stickiness of the sand is negligible, so that the grains remain at least several times smaller than the opening through which they pour. That should be true as long as the inside of the hourglass remains dry, a reasonable assumption for indoor spaces in winter in your location.

  8. Maybe this is a kind of paradox, but in an experiment such as this one, our experimental equations do not contain a term for “time” (correct me if I am wrong). Then how do we justify our claim that we “measure time?”

    The way I understand it, in this experiment, we are simply counting changes in direction of one object (flipping the sand timer) by using the direction changes of our unit (oscillations of the timer in the Android phone).

    We are comparing direction changes in our unit and the direction changes in our measured object? How does “time” enter into this process?

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