Students in Lowell, Mass., are petitioning for the right to vote in their town. They argue that voting is an integral part of the civics they’re learning in high school, and they should be able to learn about, and vote on, issues in the communities where they grew up, and not have to wait until they’re in college.
Their campaign gained support of local politicians, cause the passage of a state law allowing the town to lower the voting age and even got a constitutional law professor (Lawrence Tribe) to write a letter supporting their case to the Massachusetts’ Secretary of State. They’re now organizing a ballot initiative for 2013.
Using stopwatches, we measured the time it took for the tennis ball to fall 5.3 meters. Some of the individual measurements were off by over 30%, but the average time measured was only off by 7%.
I did a little exercise at the start of my high-school physics class today that introduced different types of experimental error. We’re starting the second quarter now and it’s time for their lab reports to including more discussion about potential sources of error, how they might fix some of them, and what they might mean.
One of the stairwells just outside the physics classroom wraps around nicely, so students could stand on the steps and, using stopwatches, time it as I dropped a tennis ball 5.3 meters, from the top banister to the floor below.
Students' measured falling times (in seconds).
Random and Reading Errors
They had a variety of stopwatches, including a number of phones, at least one wristwatch, and a few of the classroom stopwatches that I had on hand. Some devices could do readings to one hundredth of a second, while others could only do tenths of a second. So you can see that there is some error just due to how detailed the measuring device can be read. We’ll call this the reading error. If the best value your stopwatch gives you is to the tenth of a second, then you have a reading error of plus or minus 0.1 seconds (±0.1 s). And you can’t do much about this other than get a better measuring device.
Another source of error is just due to random differences that will happen with every experimental trial. Maybe you were just a fraction of a second slower stopping your watch this time compared to the last. Maybe a slight gust of air slowed the balls fall when it dropped this time. This type of error is usually just called random error, and can only be reduced by taking more and more measurements.
Our combination of reading and random errors, meant that we had quite a wide range of results – ranging from a minimum time of 0.7 seconds, to a maximum of 1.2 seconds.
So what was the right answer?
Well, you can calculate the falling time if you know the distance (d) the ball fell (d = 5.3 m), and its acceleration due to gravity (g = 9.8 m/s2) using the equation:
which gives:
So while some individual measurements were off by over 30%, the average value was off by only 8%, which is a nice illustration of the phenomenon that the more measurements you take, the better your result. In fact, you can plot the improvement in the data by drawing a graph of how the average of the measurements improves with the number of measurements (n) you take.
The first measurement (1.2 s) is much higher than the calculated value, but when you incorporate the next four values in the average it undershoots the actual (calculated) value. However, as you add more and more data points into the average the measured value gets slowly closer to the calculated value.
More measurements reduce the random error, but you tend to get to a point of diminishing returns when you average just does not improve enough to make it worth the effort of taking more measurements. The graph shows the average slowly ramping up after you use five measurements. While there are statistical techniques that can help you determine how many samples are enough, you ultimately have to base you decision on how accurate you want to be and how much time and energy you want to spend on the project. Given the large range of values we have in this example, I would not want to use less than six measurements.
Systematic Error
But, as you can see from the graph, even with over a dozen measurements, the average measured value remains persistently lower than the calculated value. Why?
This is quite likely due to some systematic error in our experiment – an error you make every time you do the experiment. Systematic errors are the most interesting type of errors because they tell you that something in the way you’ve designed your experiment is faulty.
The most exciting type of systematic error would, in my opinion, be one caused by a fundamental error in your assumptions, because they challenge you to fundamentally reevaluate what you’re doing. The scientists who recently reported seeing particles moving faster than light made their discovery because there was a systematic error in their measurements – an error that may result in the rewriting of the laws of physics.
In our experiment, I calculated the time the tennis ball took to fall using the gravitational acceleration at the surface of the Earth (9.8 m/s2). One important force that I did not consider in the calculation was air resistance. Air resistance would slow down the ball every single time it was dropped. It would be a systematic error. In fact, we could use the error that shows up to actually calculate the force of the air resistance.
However, since air resistance would slow the ball down, it would take longer to hit the floor. Unfortunately, our measurements were shorter than the calculated falling time so air resistance is unlikely to explain our error. So we’re left with some error in how the experiment was done. And quite frankly, I’m not really sure what it is. I suspect it has to do with student’s reaction times – it probably took them longer to start their stopwatches when I dropped the ball than it did to stop them when the ball hit the floor – but I’m not sure. We’ll need further experiments to figure this one out.
In Conclusion
On reflection, I think I probably would have done better using a less dense ball, perhaps a styrofoam ball, that would be more affected by air resistance, so I can show how systematic errors can be useful.
Fortunately (sort of) in my demonstration I made an error in calculating the falling rate – I forgot to include the 2 under the square root sign – so I ended up with a much lower predicted falling time for the ball – which allowed me to go through a whole exercise showing the class how to use Excel’s Goal Seek function to figure out the deceleration due to air resistance.
My Excel Spreadsheet with all the data and calculations is included here.
There are quite a number of other things that I did not get into since I was trying to keep this exercise short (less than half an hour), but one key one would be using significant figures.
There are a number of good, but technical websites dealing with error analysis including this, this and this.
“It’s one of the places I’m most proud to bring people when they visit St. Louis,” commented (more or less) one of the other faculty on our field trip to the Laumeier Sculpture Park. My hope was that this trip, combined with our visit to the Leonardo Da Vinci Exhibition, would be a nice way to demonstrate that the distance between art and science isn’t so large after all.
I believe this sculpture is called, "Balance".
I required all of my students (physics and middle school science) to identify their favorite piece and sketch it. We’ll be covering forces, balance and mechanics in the coming quarter, making this part of the spark-the-imagination part of the lesson.
Also, detailed sketches are not easy. An accurate drawing requires a lot more careful observation than even taking a picture. By trying it themselves they’d get a much greater appreciation of Da Vinci.
The large pieces are quite impressive. The brightly painted metal tank combination at the top of this post just towers over everything. However, one of the neatest was carved out of one enormous piece of wood. It’s made to emulate the distinctive, ridged bark of the cottonwood tree. The artist accentuates the ridges and valleys quite elegantly, making a wonderfully warm and organic abstraction.
Looking up at the cottonwood tree sculpture.
There’s also an indoor museum (which was closed while we were there), and, apparently, pieces are added and taken away so the park is worth revisiting. The only problem is that you’re not allowed to climb on the sculptures. This is a quite understandable precaution to protect the pieces, but, as some of the students observed, the sculptures “invite” you into and onto them. It’s a stark contrast to the St. Louis City Muesum, which is designed specifically to be played on.
A thermostat that adapts, is connected, saves money, saves energy (and the environment), and looks good too? (Image from the Nest Labs website).
There is a lot of potential for the new Nestthermostat to bring modern technology into some of the the essential but mundane devices that surround us. Its importance is not in the addition of new technology (which usually means new complexity), but in how that technology can actually make life easier, help save money, and reduce our impact on the environment by saving energy. Steven Levy has an excellent article in Wired about the project.
A key thing that students should note is the large range of expertise that went into creating this device: engineers, computer scientists, venture capitalists, and artistic designers to name a few. The ability to collaborate with diverse groups is an essential skill to master.
Just because of the large savings that can be gained, thermostats have been long overdue for an overhaul. Most buildings are heated by burning fossil fuels, like natural gas or coal, or with electricity that is produced by the burning of of fossil fuels. Similarly, cooling is also usually powered by electricity. Thus every savings in energy that results from this thermostat reduces the human impact on global warming. Because the energy savings means that you pay less for energy, saving the environment in this way means that you’re also saving money.
It’s good to see projects like this one come to fruition. We can only hope that they did a good job, that this is actually a good product, and that it is successful so similar projects will follow.
How many people were alive when you were born? How many people had lived before you were born? How many people were born while you were figuring this out?
Notes (by Natasha D.,"aged" and reflected) from our visit to the DaVinci Exhibit.
Working models of Leonardo Da Vinci’s devices, and video of his sketchbook, so inspired one student that she emulated Da Vinci’s style as she took her notes during our visit to the Da Vinci Machines Exhibition. While I’d asked them to bring their notebooks, I’d not said anything about taking notes (nor is there to be a quiz afterward) so it was very nice to see this student’s efforts. The exhibition is in St. Louis at the moment, until the end of the year.
Scan of a page of the Codex de Leicester by Leonardo DaVinci. (Image via Wikimedia Commons).Flywheel using spherical weights. Constructed based on Leonardo da Vinci's drawings. Photo by Erik Möller via Wikimedia Commons.
What I liked most about the exhibit is that you can operate some of the reconstructions of flywheels, gears, pulleys, catapults, and other machines that came out of DaVinci’s notebooks.
Da Vinci did a lot with gears, inclined planes, pulleys and other combination of simple machines, so the exhibit is a nice introduction to mechanics in physics. The exhibition provides a teacher’s guide that’s useful in this regard.
It’s an excellent exhibition, especially if you spend some time playing with the machines.
A selection of (as yet) unidentified fungi from the school campus in eastern Missouri.
It’s remarkable how interest drives motivation and motivation gets things done. We’re in an intercession right now and ten students signed up with me to do a biological survey of the school grounds. With a small creek on one side, and a fairly tall ridge on the other, the school has a nice variety of biomes.
Now, to be clear, I’m not a biologist. In fact, that’s why I was so interested in the biological survey. Everything in this area is new to me. But it also means that I approached this project as a novice. Mrs. E. was nice enough to lend me a veritable library of reference books, covering everything from the wildflowers of Missouri to the amphibians of the mid-West, but she was off teaching another batch of students how to cook, so I was on my own.
All the students in the group were volunteers, but a fair chunk of them just wanted to get outside, even though it was overcast and threatening rain. To get the students more engaged I let them choose either the environment they’d like to survey, or the types of organisms they’d like to specialize in. I also gave them the option of working independently or in pairs.
The Creek
The Creek team collected a pair of amphibians. They were documented, photographed, and then released.
One pair choose to canvas the small creek that runs past the school. I’d set a minnow trap the night before to collect fish for our tank, and they hauled that in. The stream water was somewhere around 14°C, while our tank was closer to 23°C, so, to prevent the fish from going into thermal shock, we left the minnows in a bucket so it could, slowly, thermally equilibrate. They monitored the temperature change with time, and I think I’ll use their data in my physics and calculus classes.
They also collected a pair of amphibians, which we photographed and then released. They tried to catch some crawfish, but were unsuccessful, despite the fact that one of them searched for “how to catch crawfish” on their phone; unfortunately they did not have time to follow the detailed video instructions they found on the web that described, in detail, how to build a crawfish trap.
Trees and Shrubs
Collected leaf specimens PL01 and PL02.
Because of the incipient rain, we did not take our reference books out with us. Instead, we collected leaves and sketched bark patterns so we could do our floral identification later.
Berries from an (as yet) unidentified bush.
A number of students really got into that. So we have a fairly large collection, though almost all of which come from the riparian area that bounds the creek. I would have liked a broader survey, but we only had so much time.
Unidentified wildflowers.
Mushrooms
Part of our mushroom collection.
More than a few students were interested in looking for mushrooms – even one of the tree specialists came back a mushroom sample – but one student really got into it, canvasing all the dead logs from the creek, through the meadow, and up past the treeline on the side of the hill.
The underside of this fungi looks a bit like a brain coral.
And we now have quite the collection of fungi. They’re as yet unidentified, but they’re elegant bits of biota. Our fungi specialist is interested in coming back in and sketching them.
Identification
We had two hours. Not even enough time to do a complete survey, so we barely got started on identification. It will probably go slowly.
While our methods were not systematic, and our coverage of the grounds incomplete, this exercise was a good start to cataloging the local biology. I don’t know if I’ll be able to expand on the survey any time soon, but this type of project would be a great for middle school science next year when we focus more on the biological sciences, particularly on taxonomy.
Two weekends ago we lucked into some of the most beautiful fall weather for our walk-a-ton. We’re trying to replace our old playground that had become somewhat decrepit. While our objective is not quite up to the Skudeneshavn Primary School‘s standards, it’s somewhat closer than what we had before.
Most of the kids who walked around the lake were from the lower school – after all, they’re the ones who expected to “play”- but I expect it will gets a lot of use by the middle and high schoolers once it’s installed.
Some walked more than others, but everyone had a good time.
