# Gas station without pumps

## 2012 January 13

### Newton’s measurement of g

Filed under: home school — gasstationwithoutpumps @ 22:02
Tags: , , , , , , , The motor for the circular pendulum mounted on a ceiling beam, with a short string attached.

In Chapter 5 of Matter and Interactions, the authors describe an experiment that Newton did to measure $g$, the gravitational field at the surface of the Earth.  The idea is a simple one: have a pendulum moving in a circle.  The size of the circle is a function of the length of the pendulum, the speed of rotation, and $g$. So I thought it would be fun to do an experiment along those lines in physics lab today.

The first step was to make a circular pendulum that could be driven to move at a constant speed.  My idea here was to build something out of Lego, so that we could adjust the speed as needed with pulleys and belts. My son and I built such a device early in the week and put it up on one of the ceiling beams in the room where we do our physics labs.  I put it up that high so that we could have a long string for the pendulum that would be easy to measure.  I had not done any calculations, just guessed that the bead on the string would fly out as we spun the shaft.

We had quite a surprise when we tried it out with a fast-spinning motor (about  9Hz or 540rpm) and a 2m long string.  The bead hung straight below the motor, but the string bowed out in several places, forming a nice pattern of nodes and antinodes. Changing the speed of the motor changed the number of nodes and antinodes. That is where I stopped the pre-lab preparation—the rest of the lab was done with the students.

The first thing we did was rederive the formulas the motion of the mass at the end of the pendulum, using the new whiteboards I had made for the class.  The whiteboards cost under $8 each for 2 2’×3′ boards—the lumberyard sells dry-erase hardboard for$1 a square foot (less if you buy a full sheet, but that was too hard to get home on my bike), plus $1 for the cutting. The rest was the cost of the duct tape for the edge and taxes. The cone formed by the string can be described as having height $h$, base radius $R$, and hypotenuse $L$, the length of the string. The idea was to measure $L$, $R$, and the period of the pendulum $T$, and from these try to calculate $g$. We neglected air resistance (probably not a good idea) and so had just two forces acting on the mass: the gravitational attraction to the Earth $(0, -mg, 0)$ and the diagonal pull from the string with magnitude $F_s$. We can get the horizontal and vertical components of the force exerted by the string: $F_s(R/L, h/L)$. The vertical component must match the force due to the Earth as the mass is not moving vertically, so $F_s h/L=mg$, or $F_s = mgL/h$. The magnitude of horizontal component is $F_s R/L= mg R/h$. The horizontal component is what is accelerating the mass around the circle. We rederived the formula for that by describing the position of the mass in the complex plane $\Xi(t) = R e^{i\omega t}$ and taking the derivative twice: $d^2 \Xi(t) / dt^2 = -R \omega^2 e^{i\omega t}$. One of the students was very comfortable with this, the other (who had ostensibly had more calculus), had never seen $e^{i \omega}$. I gave a very brief mention of doing it with sines and cosines, but did not take the time to do the derivation in those terms. Note that the magnitude of the force is $m R\omega^2$ and it is directed towards the center of the circle, as we would expect. The angular frequency $\omega$ is just the speed in radians per second, so $\omega=2\pi/T$. Combining the two formulas for the horizontal force, we get $mgR/h = mR\omega^2 = m R 4 \pi^2 /T^2$. We can simplify this to $g=4 \pi^2 h/T^2$. If we don’t want to measure $h$,we can use Pythagoras’s Theorem to get $g= 4\pi^2 \sqrt{L^2-R^2}/T^2$. After demonstrating the unexpected behavior of the long string, I challenged the students to compute $h$ from the known value of $g$. To do the calculation we needed the period $T$. In preparation for this, I had bought a “photo interrupter” from Sparkfun and a breakout board to mount it. (Actually, I ordered 2, which was a good thing, since one of them did not work—Sparkfun is sending me a replacement). I hooked up the photo interrupter to my Arduino microprocessor, which I programmed to measure the time between falling edges of pulses (with a 20msec minimum pulse width). By letting the Lego beam on the motor pass through the gap, I could get pretty consistent measurements of the period. Because the beam is not quite centered, I believe I was getting just one end to pass through the gap. I could have made the measurement easier by having a piece of opaque tape sticking out the end of the beam to pass through the gap. I believe that the period as the motor was initially set up was 119 msec. The students computed that this should result in lifting the mass by 3.5 mm, which is a bit small to measure at the end of a 2m string! The next step was to have the students compute a period that would involve a decent value for $h$, say 20 cm. They computed it to be about 0.9 seconds, which was easily arranged by changing the pulleys on the motor and shaft. We then shortened the string to about 30 cm, and put the motor on the desk. I forget exactly what the period turned out to be: something close to 910 msec, I believe. We did not take careful notes, because we were still playing around with the setup when it was time to end the class (one student’s mother came to fetch him, another had to go off to improv class, and I had a meeting with 4 grad students and an AP bio teacher about a bioinformatics lesson the students are going to do for his classes in 2 weeks). Why were we still fussing around? Well, when we turned on the motor the pendulum mass did indeed move out into a big circle, but it didn’t stay there! Instead the circle slowly shrank until the bead was hanging almost straight down, then grew again to a big circle, then shrank again, and so on. This was the second, even less anticipated result from the lab. I still don’t have a good explanation for it. Am I getting some sort of beating between the natural period of the pendulum and the forced rotation? I think I’ll have to play around with the setup this weekend to see the effect of different variations. Some things I plan to play with include • using a heavy weight for the mass, instead of a tiny one. Originally, I had thought the I would need to allow the mass to pass through the gap in the photo interrupter, and I didn’t want to risk damaging it. But now I see that I can measure the period using the beam, so the mass can be larger. • adjusting the length of the string, to change the natural period of the pendulum. • adjusting the speed of rotation. It would be nice to have a real slow rotation, so that I could do a decent measurement with a long string. Question for the physics people who read my blog: have any of you done the circular pendulum experiment in class? Are there some things I should have known about to make this work better? Do you have an explanation for why I got such a varying amplitude in the size of the circle with the short pendulum? Will the long string with a fairly fast motor always produce nice nodes and antinodes, or did I get lucky on the two speeds I tried? ## 12 Comments » 1. I don’t know that the motor is really necessary. When we do this, I attach the string to the ceiling, and have a 2.8 m pendulum or so. I can usually just give it a short push, and it will move in a circle for a long enough time for us to get a very good measurement of the period. We also don’t use the photogate—we just time the time it takes to cross a line on a piece of paper underneath, and measure out the time for 10 periods. But I like the photogate idea a lot. Comment by John Burk — 2012 January 14 @ 06:11 • In addition to the period, you need to measure some parameter that gets you the angle of the cone. Do you measure the length of the string and the radius of the circle as I planned? the height that the mass lifts by? the angle directly? When I was planning the lab, I thought I would need to keep adding energy to keep the period constant (I had already planned to use a very light pendulum). With a heavier pendulum, I can see that might not be necessary, though the tiny 1cm gap of the photogate I have would be hard to hit with the tip of a decaying pendulum. The motorized approach allows me to measure the period at a more fixed position (near the motor). I think that my first experiment will be to use a much larger mass, since the tiny bead I was using is not much heavier than the string. Comment by gasstationwithoutpumps — 2012 January 14 @ 08:47 2. In addition to the period, you need to measure some parameter that gets you the angle of the cone. Do you measure the length of the string and the radius of the circle as I planned? the height that the mass lifts by? the angle directly? Maybe I’m missing something, but the typical answer is… no, you don’t need anything else. You make sure the pendulum is very long compared to the amplitude of the swing (or the radius of the circle), then you take the small-angle approximation, and then the period of your conical pendulum will be the same as the period of an ordinary pendulum, t=2\pi \sqrt{L/g} Comment by secretseasons — 2012 January 14 @ 09:58 • If I understand correctly, the small-angle approximation is that $h \approx L$, that is, that the vertical height is close to the length of the string, which is probably good enough for a very long pendulum. Comment by gasstationwithoutpumps — 2012 January 14 @ 10:22 • Maybe back up a step — why are you measuring g with a conical pendulum? Not that there’s anything wrong with that, but did you have a specific reason to do it that way, as opposed to just a “ordinary” pendulum? Comment by secretseasons — 2012 January 14 @ 10:26 • For the simple reason that the conical pendulum is what Newton is said to have used (and one of the students is a history buff). Also, with the circular pendulum motion we don’t need to make the small-angle approximation and can compute the period analytically. Mathematically, the circular pendulum is simpler than the simple pendulum. A more honest answer is that the book explains circular pendulums but only does simple pendulums as an exercise at the end of Chapter 4. I did not assign that exercise, so we did not derive the period of a simple pendulum. Perhaps I should go back and have students do at least the math part of that exercise. Comment by gasstationwithoutpumps — 2012 January 14 @ 10:41 • Oops, I did assign problem 4P89, we just haven’t done it yet! So next week’s lab: simple pendulums! (and finish up the circular pendulum lab). Comment by gasstationwithoutpumps — 2012 January 14 @ 11:33 • Also, with the circular pendulum motion we don’t need to make the small-angle approximation and can compute the period analytically. Mathematically, the circular pendulum is simpler than the simple pendulum. It seems to me that if I view the pendulum from above and think of x and y like the axes of a Lissajous figure then the simple pendulum is just one axis of the 2-d motion and from that perspective I’d consider the simple pendulum to be, well, simpler. But I guess that’s because I wouldn’t expect the e^i\omegat treatment that you’re using, so maybe in your formalism, you’re right. But I would expect all high school physics students to take away the lesson that (for small amplitude, anyway), the period does not depend on the amplitude of the oscillation. This is what makes them great timekeepers (and why the motor seems unnecessary). So I worry that this treatment misses the point, in a way, even if it is more rigorous (and perhaps in some sense, more correct). Comment by secretseasons — 2012 January 14 @ 12:37 3. First test: heavier weight. With a large weight I get an almost ellipsoidal motion of the weight, with the major axis of the ellipse gradually rotating. That looks like a coupling between the driven circular motion and the natural frequency of the pendulum. I need a long enough pendulum and I need to drive the pendulum fast enough that $h < L$. If $h << L$, then $R$ will be large and the weight will be flying around in a possibly dangerous way (assuming the motor has enough power to keep the pendulum spinning). If $h$ is only slightly smaller than $L$, then the natural frequency of the pendulum and the driven frequency will be similar, and I'll see the rotating ellipse (unless I can get the initial starting velocity just right). If I use an undriven circular pendulum, I'll need to find a way to measure the period with the photogate, which might be hard with the mass moving in a spiral. A photogate with larger gap (like the$10 ones that use a modulated IR beam and can span 20m) would make this much easier, though the beam spread may be a problem, since they are intended for detecting big objects, like people or cars. I think the simple pendulum is easier here, since the mass passes through the rest position repeatedly, so a small photogate is not a problem.

Comment by gasstationwithoutpumps — 2012 January 14 @ 12:34

4. In response to secretseasons, I agree that the time-keeping nature of pendulums is an important thing for students to learn, but that requires measuring each swing of the pendulum separately (averaging many swings presupposes the constancy). I thought that the 10µsec timing resolution of the cheap optical gate setup would be great for this, but I can’t seem to hit that tiny 1cm gap consistently, even with a simple pendulum.

For just measuring $g$, averaging over many periods of a circular pendulum should be fine, and I’ll see if I can get that setup to work using just a stopwatch. With a long enough pendulum and enough periods, we’ll be able to make the time long enough that crude measurement (±200msec) on the stopwatch won’t matter much. I still want to find a way to use the Arduino to time the simple pendulum, though. I suppose I could try interfacing the magnetometer chip and detecting a magnetic weight, but that can’t take samples any faster than 12msec, so is nowhere near the timing resolution of the simpler optical gate.

Comment by gasstationwithoutpumps — 2012 January 14 @ 13:58

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