In today’s lab we derived the formula for the period of a simple pendulum (assuming the small-angle approximation), , then measured both circular and simple pendulums. For the circular pendulum we measured the radius of the cone on the first orbit and the last orbit, the length of the string (the slant height of the cone), and approximated the period by timing 10 or 20 periods and dividing. For the simple pendulum, we used the photogate setup described in More on pendulums, to get very precise and repeatable measurements of the period. The hardest part for us was measuring the length of the pendulums, since the center of mass for the bob was not obvious and the exact position of the pivot was not obvious—these uncertainties probably resulted in length measurements being ±5mm, making a large contribution to inaccuracy.

Here is a table of the measurements (and calculated g) we made for the circular pendulum:

Length cm | radius cm | num orbits | period sec | g cm/sec^2 |
---|---|---|---|---|

212.4 | 48.6–46.6 | 10 | 2.90 | 970.8–972.6 |

212.4 | 38–52.4 | 20 | 2.601 | 959.8–974.7 |

161.5 | 58–60.5 | 20 | 2.501 | 938.7–984.2 |

The range of estimates for g is larger than I would like. I think that the decay of the oscillation of the pendulum makes quite a difference. The average of all the estimates of g is 967 gm/sec^2, which is rather low.

And for the simple pendulum:

Length cm | num ticks | mean period sec | standard deviation | g cm/sec^2 |
---|---|---|---|---|

207.2 | 47 | 2.8958 | 0.0050 | 975.4 |

171.3 | 74 | 2.6272 | 0.0065 | 979.8 |

95.5 | 89 | 1.9565 | 0.0025 | 984.9 |

54.7 | 58 | 1.4809 | 0.0042 | 984.7 |

28.7 | 44 | 1.0730 | 0.0019 | 984.0 |

The pendulum ticked reliably for quite a while, and the periods were remarkably consistent. The estimates of g from the simple pendulum are good to about 0.5%, which is the limitation of accuracy on our pendulum length measurements and close to the limit of the accuracy of the small-angle approximation. The average of the 5 measurements looks good to about 0.2%, which seems pretty good to me, since we certainly weren’t measuring the lengths that accurately.

I looked up the gravitational field in Santa Cruz on Wolfram Alpha’s gravitational fields widget:

total field | 9.7995 m/s^2 (meters per second squared) angular deviation from local vertical | 0.00322° (degrees) down component | 9.79945 m/s^2 (meters per second squared) west component | 3.4×10^-4 m/s^2 (meters per second squared) south component | 0.0316 m/s^2 (meters per second squared) (based on EGM2008 12th order model; 11 meters above sea level)

While the lab was running, one of the students wrote a Python script (using numpy for mean and standard deviation) to read the data and compute the numbers in the table. We could have talked directly to the Arduino, but it was simpler to cut the numbers from the Arduino serial monitor and paste them into a file for the script to read. That allowed us to keep the Arduino running throughout, and just cut and paste the good numbers, discarding the junk from starting or stopping the pendulum.

I’m quite pleased with the photogate setup, which was very simple to build and worked reliably during the experiment. Crudely wrapping tape around the string made a lumpy opaque object, whose rotation probably contributed to the standard deviation of the period—having a smoother cylinder for the optical blocker would probably make the period measurement much more consistent. But that would not improve the mean estimates much since errors in adjacent period measurements cancel. I believe that our mean periods are much more accurate than the standard deviations suggests, with errors less than 1 per thousand.

I had to make one change in the Arduino code during the lab to accommodate all the different pendulum lengths—I had a dead time before recognizing the next pulse, to prevent getting 2 pulses per period as the string passed through the beam twice. I started with a dead time of 1 second, which as a bit too long for the smallest pendulum. Reducing the dead time to 500 msec for that pendulum made it count reliably. Note that for the 2nd and 3rd pendulum, we measured for about 3 minutes without a bad time measurement, and could have gone longer if we had had the patience.