Math/Science with 7th graders (2): Pendulums

I decided to introduce pendulums this week and see what Math and Science it throws up. I found a small section on pendulums as part of the chapter on measurement of time (p.g.184-186 7th Std) and decided to pursue it. Any measurements or activity take a lot of time and I decided to let it take its time, find the doubts about different aspects and see if we can talk about them some more.

On the first day we spent some time figuring out how to go about our experiments. We collected stones, found some thread and tied it to a meter long scale that would rest across two tables. It gave us around 70 cm from the floor and that determined the limit of our experiments. We started our experiments with a 60 cm long thread.

I clarified the goal of the measurement i.e. a period which is the time the bob takes to go from one end to the other and get back. I asked them to make guesses in the experiments to exercise their intuition and explain why they picked up these numbers to explore their assumptions. Its interesting that everyone assumes that the period of a pendulum will be one second irrespective of the length...

Kids	Small Amplitude	Large Amplitude	Large Weight
Arc	2	2	1.3
Sha	2	2	1.2
Var	1	1.9	1
Sub	1	2.5	1
Pri	2	2.5	1.2
Actual	1.6	1.62	1.56

(**All measurements and guesses in seconds)

The experiments we were able to complete on the first day were the impact on the amplitude on the period and the weight of the bob on the period. They had anticipated that the larger the amplitude the longer the time period, since longer the bob needs to travel. They had also anticipated that a heavier bob will have a shorter period as it has more 'strength'.

We did the measurements using my digital watch. Apparently, it can count to a resolution of 1/1000 s, though I doubt if our triggering accuracy was any better than 1/10 s. I made the first couple of measurements, but following that most of the measurements were made by the kids themselves. To squeeze in division of decimals I suggested that we could 5 periods, the logic being that the amplitude of the oscillations may change significantly beyond that. The children throughly enjoyed the discussions, guessing and even the decimal calculations that came as a result of their experiments.

Its a effort to tie thread around a stone and it sometimes breaks or loosens up. The kids really winded up around a big stone for quite some time till it became our staple for the next day measurement as well.

The second day we reiterated what we learnt namely, bob weight and amplitude does not matter. Now we went after the length to see what we get. Again we went for guesses.

Name	50 cm	40 cm	31 cm	11 cm
Arc	1.6	1.3	1.25	0.5
Sha	1.7	1.2	1.2	0.4
Var	1.6	1.4	1.25	0.52
Shu	1.3	1.3	1.25	0.5
Pri	1.4	1.3	1.25	0.6
Sanjeev	1.45	1.35	1.18	0.7
Actual	1.4885	1.297	1.22	0.839

As we had concluded that the amplitude does not have much of an effect (not true unless it is 'small') we decided to count 10 periods instead of 5 to make calculations easier. We were further able to refresh our decimals by talking about how close the guesses were to the Actual and who was closer. I also participated in the guessing and didn't cheat or use calculations, just gut the way the kids were doing. It was fun to play on a level playing field.

This class went faster and I split a graph sheet to see if we could plot it and see that it did not lie on a straight line. Also I wanted to re-emphasize the decimal numbers on the graph sheet with different scales.

Plot from measurements

Ideal Plot from T=2*pi*sqrt(l/g)

I plot the measurements we got and also the plot with the ideal equation. I noticed that with the ideal curves the error for the measurement with the smaller length was quite a bit. I realized that our stone was not only heavy but large and we needed to take the CG of the stone into account. I wasn't sure how I was going to convey it to my students as this fell in an area where my language skills may prove insufficient.

I started thinking about alternatives to the large stone and thought of where I could get hold of metal bolts. I went to the local two wheeler repair shop and I poked around in his waste/spare and found a goldmine of washers, bolts, etc.

I also wondered how to convey the non-linear nature of the curve to the kids...

I wasn't sure if pendulums were very useful anymore and decided to look around. Turns out the pendulums with temperature compensation were used as a standard of time till not so long ago and they did keep time for humanity for a good part of three centuries. I found a bunch of videos on youtube both on things that pendulums are useful for e.g. ballistic pendulum and things that I thought would be of interest to the kids and what they would analyze.

The first video was a series of pendulums of various lengths that created very interesting patterns. The second was a video of a coupled pendulum that explains the idea of resonance. A cute trick with the idea of pendulum that takes intuition a few seconds to catch up. I found a home made Barton's pendulum that explained how the coupled pendulums work. I took a snippet from an MIT class where the teacher sits on his 15 kg pendulum to show that weight indeed doesn't matter for the period of the pendulum. I also had one of Walter talking about potential, kinetic energy conversion and the experiment where he puts his face on the line to make his point. The newton's cradle also used the pendulum in a different way than we had seen in the class. I also took a video of a science lab with the experiment on the ballistic pendulum and a couple of people who were trying to make physical swings work as coupled pendulums! I also refreshed my idea of the Foucault's pendulum, but decided only to mention it rather than show it.

On the third day we went through all the videos and tried out the trick with the pendulum to start with. The only video that was a flop was the science experiment where the person was describing how to use the ballistic pendulum (science teachers can make a gun being shot boring)!

We talked about what in these we understood immediately, this was the patterns created by the series of pendulums of various lengths. The children knew that the lengths would determine the period and that things would not be in sync, they were now not looking at science, but at the art and beauty created in the exercise. They want to build one of this.

Armed with the new found bolts, washers we ventured on trying the 'cute trick' even as we had just finished watching the video. We spent some time building the coupled pendulum's that had really caught their attention.

We started having a customary circle time to talk about what we had learnt so far at the beginning of a class and in the fourth day we started talking about why the pendulum swings at all, what are the forces on it and what is the speed of the pendulum at different places of the period. To help them remember how it takes time to speed up I had the most energetic kid ran back and forth in the room (I told him it only counts if he doesn't bump into anything to stop). He said that he was fastest when he was in the middle of the room, everyone agreed to his assessment.

We also talked about how the period varies with the length. They all remembered that it decreases with the length. I asked them how much the period of a pendulum of 60 cm was, they remembered 1.6s. I asked them how much the period of a 15 cm pendulum was they said 1/4th. Of course I do not let go of any opportunity in which they need to do any calculations with fractions or decimals and boy this had both! They came back with 0.4 s. I asked them how confident they were and then asked them to try to remember what happened when we tried to measure the 11 cm pendulum and compare it to 0.4 s. They were now convinced that they had no clue and wanted to do the experiment. Yes, it turned out well with the period only going to 0.8 s. I told them that they now knew enough Math to understand how this works. 

We had already gone over powers of a number like 5^4 and I talked about the two special ones squares being area and cubes being volume i.e. if the side of a square was 3 then its area was 3^2=9. I then talked about the complementary question (like subtraction for addition and division for multiplication as I explained) if you had the area of a square what would be one side of it. This helped me introduce a square root. I asked them to now try to figure out by how much the period will change if the period is related to the square root of the length. Yes, it took them a little time, but they arrived at 1/2 and then correlated it to what they had observed.

I shared that we had done something special this week, they agreed (perhaps, they didn't have much choice :)) and that we should showcase the week by building something for the teachers for the meeting the next day.

The fifth day was rained out and only three kids came. One of who had been sick for most of the week and had only come on the 4th day. There were very few kids at school and taking classes was difficult so an independence day movie was played for the older kids. The kids got out with less than an hour left before the teachers meeting. We found a stand that was used as a screen to do puppet shows and I carried it with the students to the staff room. Then we sat down and planned about what we could do in the time we had. They decided to do both the coupled pendulum and the Barton's pendulum to explain that resonance is important for the coupled pendulum. I pretty much left them to themselves and once they were done they also gave a demo of what they had made that I recorded these videos for the teachers.

Their effort helped have a good discussion at the meeting among the teachers... Anyway, I had a fun week, it was challenging to let go of what they should learn in a certain time, but in the end they were able something about a physical phenomenon, build stuff that made them question how things work, see math as a way of estimating, guessing and analyzing physical phenomenon, work with decimals and put the numbers and decimals on a graph and think about fractions visually.

We also played a pizza party fraction game on the fourth day, but that's for a different blog post. For now I leave you with a short video of a kid trying to demo the Barton's Pendulum experiment.