The Physics
Hypertextbook
Opus in profectus

# Curve Fitting

## Problems

### practice

1. simple-pendulum.txt
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?
2. vostok.txt
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.
Coumns:
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.

Questions…

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?
3. pizza.txt
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?

### statistical

#### two-column data sets

1. aerodynamic-drag.txt
In this experiment students measured the aerodynamic drag on a weighted party balloon falling at different speeds. How does speed affect aerodynamic drag?
2. circular-motion.txt
In this experiment a mass on the end of a string under constant tension was swung in a circle. What is the relationship between the speed of the mass and the radius of the string when the centripetal force is constant?
3. constant-force.txt
In this experiment different masses were subject to the same force and their accelerations recorded. What effect does mass have on acceleration when force is constant?
4. falling-bodies.txt
In this experiment different masses were dropped and their accelerations recorded. What effect does mass have on the acceleration due to gravity?
5. hubble-law.txt
The data gathered in this classic Twentieth Century scientific paper show the relation between the distance to several nearby galaxies and their radial velocities (that is, the component of their velocities away from the Milky Way). What is the relation between these two quantities and what does it say about the universe as a whole? This observation was described in 1929 by the American astronomer Edwin Hubble (1889–1953) and is now known as Hubble's law. Source:  A relation between distance and radial velocity among extra-galactic nebulae. Edwin Hubble. Proceedings of the National Academy of Sciences. Vol. 15 No. 3 (1929): 168–173.
6. milk-freshness.txt
The following data were taken from a milk carton sold in North Carolina. How is shelf life affected by storage temperature? Source: Greenler, Robert. Chasing the Rainbow. Milwaukee, WI: Elton-Wolf, 2000: 140.
7. moore-law.txt
This data set shows the number of switches in a computer for various years in the Twentieth Century. How has the processing capacity of computers changed with time? This relationship, in a slightly modified form, was first described by Gordon Moore (1929–0000), the founder of Intel, in 1964 and is now known as Moore's law. Source: IBM Gallery of Science & Art, 590 Madison Avenue, Second Floor, New York, NY 10022 (July 1992).
8. resonance-tube.txt
In this experiment various tuning forks of known frequency were held above a resonance tube, which was used to determine the wavelength of the sound emitted. How does frequency affect wavelength?
9. dimmer-switch.txt
In this experiment, a student placed a light meter 30 cm away from of a light bulb attached to a dimmer switch. How does the voltage delivered to a light bulb by a dimmer switch affect its illuminance? (Illuminance is related to brightness. In simple terms, illuminance is objective while brightness is subjective.)

#### multi-column data sets

1. computer-temperatures.txt
Some computers are equipped with temperatures sensors that monitor various components and activate fans. How does temperature evolve in the parts of a computer after it has been turned on?
Columns:
1. Running time (minutes)
2. CPU A Die (℃)
3. CPU B Die (℃)
4. Memory Controller Heatsink (℃)
5. Drive Bay (℃)
6. "Smart Disk" Hard Drive (℃)
2. dulong-petit-law.txt
How does specific heat vary with atomic mass? This relationship was first described in 1819 by the French physicists Pierre Louis Dulong (1785–1838) and Aléxis Thérèse Petit (1791–1820) and is now known as the Dulong–Petit law.
Columns:
1. Element name
2. Atomic mass (atomic mass units)
3. Specific heat capacity (J/kg K)
Source: CRC Handbook of Chemistry and Physics. 75th Edition. Boca Raton, FL: CRC Press, 1995.
3. foot-on-the-gas.txt
Does the price of gasoline affect the way Americans buy cars? Of course it does. Why else would I ask you? The real question is, how are gasoline prices and consumer auto purchases related?
Columns:
1. Model year
2. Average gasoline price per gallon (inflation-adjusted 1993 dollars)
3. Average engine power of cars in each model year (horsepower)
4. Average fuel economy of cars in model year (miles per gallon)
Source: After 20 Years, America's Foot Is Still on the Gas. Matthew L. Wald. New York Times (17 October 1993). Original Source: EPA, DOE, and Bureau of Labor Statistics.
4. solar-observations.txt
Heinrich Schwabe (1789–1875) was a Nineteenth Century astronomer searching for a planet nearer to the sun than Mercury. He never found this inframercurial planet, but his long term observations of sunspots lead to an important discovery. What was it?
Columns:
1. Year
2. Number of sunspot clusters
3. Days when no sunspots were observed
4. Cloud-free observation days
Source: Sonnen-Beobachtungen im Jahre 1843. Heinrich Schwabe. Astronomische Nachrichten. Vol. 20 No. 495 (1844): 233–236; Kosmos: Entwurf einer physischen Weltbeschreibung, Dritter Band. Alexander von Humboldt. Tübingen, Germany: J.G. Cotta'sche Verlag (1850): 402.
5. x-rays.txt
X-rays can be produced by bombarding stationary atoms with a beam of fast moving electrons. The energy of the emerging x-rays peaks around several values, two of which are included in the accompanying text file. x-rays with these peak values are called characteristic x-rays since they can be used to identify elements. How are energy and atomic number related for characteristic x-rays?
Columns:
1. Atomic number
2. Element
3. 1 (eV)
4. 2 (eV)
Source: National Institute of Standards and Technology (NIST). X-ray Transition Energies Database (2005).

#### two-column data sets that require some sort of transformation

1. free-fall.txt
In this experiment, the displacement of a falling object was recorded every sixtieth of a second using a device called a ticker timer. Convert the time units from sixtieths of a second to seconds and then determine the relationship between distance and time during free fall.
2. inertia.txt
In this experiment, the displacement of a cart moving in the absence of a net external force was recorded every sixtieth of a second using a device called a ticker timer. Convert the time units from sixtieths of a second to seconds and then determine the relationship between distance and time in the absence of a net external force.
3. spherical-lens.txt
In this experiment, the in-focus image distance was determined for an object placed at various distances in front of a magnifying glass. The important quantity for optometrists, however, is the reciprocal of distance in meters — a quantity called vergence. Calculate the vergences and then determine the relationship between image vergence and object vergence. (The unit of vergence is the diopter which is equal to an inverse meter [D = m−1].)

#### multi-column data sets that require some sort of transformation

1. kepler-law.txt
This data set shows the mean orbital radius (in astronomical units) and orbital period (in days or years) of each of the nine planets. Convert the radii to meters, convert the periods to seconds, and then determine the relationship between radius and period. This relationship was discovered in 1619 by the German mathematician Johannes Kepler (1571–1630) and is now known as Kepler's third law of planetary motion. He came to this conclusion 18 years after receiving the data. Given the tools of contemporary analysis, which Kepler did not have access to, you should be able to arrive at a conclusion in under five minutes.
Columns:
1. Planet name
2. Mean orbital radius in astronomical units (One astronomical unit is the average distance from the Earth to the sun.)
3. Orbital period (Note the change in units between the inner and outer planets.)
Source: CRC Handbook of Chemistry and Physics. 75th Edition. Boca Raton, FL: CRC Press, 1995.
2. track-events.txt
This file gives the world record times for eight track events.
1. Calculate the speed of each runner.
2. Determine…
1. the effect of gender on speed.
2. the effect of distance on speed.
Columns:
1. Event (length in meters)
2. Men's record times (seconds)
3. Women's record times (seconds)
Source: International Association of Athletics Federations (20 August 2010).
3. climate-and-geography.txt
What effect does latitude have on average yearly temperature? Before you attempt this question you must compensate for the variation in altitude of the different data points. Temperature decreases 6.5 ℃ for every 1,000 m of altitude. Calculate the altitude adjusted temperature for the cities in this data set. If you have done this correctly you will see the correlation of the data set increase. Then perform a curve fit and answer the question.
Columns:
1. City name
2. Latitude (degrees: positive-north, negative-south)
3. Altitude of recording station (meters)
4. Average yearly temperature (degree celsius)
Source: worldclimate.com