The Physics
Opus in profectus

Curve Fitting

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practice problem 1

In this experiment, simple pendulums of different lengths were constructed and their periods measured. This data set looks linear, but is it really? What effect does length have on the period of a simple pendulum?


Answer it.

practice problem 2

Snow rarely gets a chance to melt in Antarctica, even in the summer when the sun never sets. In the interior of the continent, the temperature of the air hasn't been above the freezing point of water in any significant way for the last 900,000 years. The snow that falls there accumulates and accumulates and accumulates until it compresses into rock solid ice — up to 4.5 km thick in some regions. Since the snow that falls is originally fluffy with air, the ice that eventually forms still holds remnants of this air — very, very old air. By examining the isotopic composition of the gases in carefully extracted cores of this ice we can learn things about the past climatic conditions on earth. By extension we might also predict some things about the climate of the future.
  1. Age of air (years before present)
  2. Temperature anomaly with respect to the mean recent time value (℃)
  3. Carbon dioxide concentration (ppm)
  4. Dust concentration (ppm)
Source: Adapted from Petit, et al. 1999.


  1. CO2
    1. Construct a set of overlapping time series graphs for CO2 concentration and temperature anomaly.
    2. Construct a scatter plot of temperature anomaly vs. CO2 concentration.
    3. How are atmospheric carbon dioxide concentration and temperature anomaly related?
    4. What temperature anomaly might one expect given current atmospheric CO2 levels?
  2. Dust
    1. Construct a set of overlapping time series graphs for dust and temperature anomaly.
    2. Construct a scatter plot of temperature anomaly vs. dust concentration.
    3. How are atmospheric dust concentration and temperature anomaly related?
    4. What global average temperature anomaly might one expect from exceptionally high levels of atmospheric dust?


  1. CO2
    1. Here are the overlapping time series graphs. The data show a definite correlation. The two quantities go up and down in near synchrony.

    2. Here's the scatter plot of the two time-varying quantities plotted against one another. The data forms a dense cloud that is roughly oval shaped. The best fit line slices nicely through the data.

    3. Temperature varies linearly with atmospheric carbon dioxide concentration. Low CO2 levels go with a cooler climate and high CO2 levels go with a warmer climate.
    4. What does our linear regression analysis predict given current carbon dioxide levels of about 400 ppm?
      y =  mx + b
      y =  (0.0908 ℃/ppm)(400 ppm) − 25.23 ℃
      y =  +11 ℃

      The current consensus among working climate scientists is that the globe will warm +5 ℃ on average over the course of the Twenty-first Century. The increase is expected to be smaller than average near the equator and greater than average near the poles. Since the Vostok ice cores were collected in Antarctica, our prediction of approximately +10 ℃ is right in line with those made by more sophisticated means.

      Correlation is not causation, however. Graphs like those used in this problem cannot tell us whether carbon dioxide affects temperature, temperature affects carbon dioxide, or some third factor is affecting both. We need a theoretical model that describes which way the cause and effect work. That model is described in more detail in the section of this book that deals with heat transfer by radiation.

      Carbon dioxide is a greenhouse gas. Its role in atmospheric thermodynamics is much like the glass in a greenhouse. It is transparent to visible light, but not to infrared. Visible light easily punches through the atmosphere. It is absorbed by the ground and then reradiated as infrared. The infrared is partly blocked by the atmosphere and has a hard time escaping out into space. This little delay keeps the Earth comfortably warm. Water vapor, carbon dioxide, methane, and other gases have been shown to play a significant role in this process. They all interact with infrared radiation. These properties have been measured in tabletop laboratory experiments that had no direct connection to climatology.

      Atmospheric carbon dioxide levels have increased steadily over the past 100 to 150 years. This is due to the burning of coal, petroleum, and natural gas as well as deforestation and other changes in land use associated with the Industrial Revolution. During this same time period, average global temperatures have been generally increasing and there is no reason to believe that this trend will quit anytime soon. Climate models all show that as long as CO2 concentrations stay somewhere around their turn of the Twenty-first Century levels, global temperatures will continue to increase for the next 100 years. This conclusion is based on solid scientific reasoning and is regarded by nearly all climate scientists as valid. The scientific questions that remain unanswered are: how can we increase the precision and reliability of our global climate predictions and what effect will the inevitable changes have on life as we know it? The question of what is to be done about this is is a political, not scientific, question.

  2. Dust
    1. Here are the overlapping time series graphs. The correlation with temperature is not as evident for dust as it was for CO2 since the changes are in the opposite direction. High levels of dust seem to correlate with low temperatures.

    2. The relation between dust and temperature becomes more evident when graphed as a scatter plot. It appears that temperature decreases with increasing dust, but this decrease levels off after a while. The relationship appears to have a horizontal asymptote like an exponential approach function.

    3. Atmospheric dust correlates to a negative temperature anomaly that exponentially approaches a minimum value.
    4. As dust concentration increases, the exponential term of our curve fitting function approaches zero leaving the constant term to stand by itself. This is the limit or asymptote of our curve fit.
      y =  aebx − c
      ymin =  − c = −7.6 ℃

      So what is going on here? Does dust cause temperatures to drop or do low temperatures make the dust fly? The answer to this question is not so simple. Here we have a real problem of determining causation.

      Water vapor, carbon dioxide, methane, and the other greenhouse gases absorb and reradiate infrared radiation because the natural vibrational frequencies of these molecules lie in the infrared part of the electromagnetic spectrum. They don't vibrate at visible light frequencies, which is why they are all transparent to the light that we see with our eyes.

      Dust particles are large compared to molecules. They are also larger than a wavelength of infrared or visible light. Because of this, they are essentially obstacles to infrared and visible light waves. When the sky is full of dust, light has a hard time making it down to the ground. The dust particles scatter a great deal of solar energy back into space and the Earth is a little bit cooler as a result. Large volcanoes can spread so much dust into the atmosphere that they sometimes have a measurable effect on climate. The 1991 eruption of Mount Pinatubo in the Philippines caused a globally averaged temperature anomaly of −0.3 ℃ that lasted two years.

      Well actually, that last statement isn't quite right. Mount Pinatubo did spread a lot of ash around the globe (resulting in fantastic sunsets), but the primary mechanism by which it cooled the Earth was through the injection of sulfur dioxide into the stratosphere. When SO2 gets together with H2O the result is an aerosol of H2SO4 — sulfuric acid. Aerosols have a greater cooling effect on climate than dust. (They are also notoriously hard to describe in climate models.)

      Now let's examine the reverse causation. A cool climate is a dry one. When temperatures are low, water has a hard time evaporating. Less water vapor means less rain. Less rain means more dust. Maybe dust levels are high when the climate is cool is because a cool climate leads to more dust. If that's the case, then the last part of this problem may not be a question that can be answered with this data set.

      If dust causes cooling, then the coldest dust could ever make the Earth would be around −8 ℃ lower than it is now. If cooling causes dust, then all we can say is that there has never been a temperature anomaly lower than −8 ℃ in the last 400,000 years and every time the Earth has gotten this cold it's been very dusty.

practice problem 3

An Internet search engine was queried for pizzerias in the area centered on Brooklyn College. The name, phone number, and address were recorded for the first six hits. How is the building number of a pizzeria in Brooklyn affected by the last four digits of its phone number?


Answer it.