College Physics Laboratory - Autumn Project Guide

Our College Physics Laboratory project for Autumn is to completely analyze that experiment. In the process, we will look at the linear motion of the puck's center of mass as well as its rotational motion, and investigate the role friction played in the experiment. We will also learn the importance of repeated measurements for quantifying error, and use a variety of plotting and statistical techniques for data analysis. Finally, we will read several historically important papers in physics in order to learn how good experimental reports are constructed. Your final report will be a culmination of this entire endeavor.

Here are two views of the puck:

It's mass is 65.56 grams, and you can measure its dimensions from the images above. Notice the solid white line on the left of the left-hand image and the 4 white dots opposite it. We will refer to the latter as the "4 dot" line and it will be useful as we measure the rotation of the puck.

About the Video

The video recording was separated into individual frames. Each frame was de-interlaced; an NTSC frame is made up of two fields, one with the even-numbered lines and one with the odd, and de-interlacing is the process of compositing them into a single image. Each image was then de-noised, cropped, brightened and over-sharpened to aid in data recording. Other than this thorough "washing", the video information was not altered in any way.

It is notoriously difficult to perform such an experiment so that the individual runs are identical; there are small differences in where the puck was started, whether it had a slight push or not and how much it may have wobbled in the track. The data sets, therefore, do not look exactly the same nor do they all have the same number of images. Your data set will be assigned to you at random and will contain from 72 to 92 images.

Taking Data - Overview

A Java applet (below) will allow you to to take all of the required data directly from the images. You will enter your data set number in the first box, and after you push the "Enter" key, the program will load the images from the server. You may not see an image until you push the play button (below).

If you are using a dial-up connection to the Internet, this could take some time; each data set contains an average of 80 images, each an average of 40 Kb in size. This means the download will average over 3 megabytes, which at a nominal 56 Kbits/second will take upwards of 8 minutes or longer, assuming error-free transmission.

The first point you click is called "Point 1", and the second one you click is called "Point 2". As you click successive points, they will alternate between Point 1 and Point 2. Each time you click on a Point 2, the program draw a red line between the points, and will also provide you with the distance (in pixels) between the points, and the angle between the vector which starts at Point 1 and ends at Point 2, and the positive x axis (in degrees, between 0 and 360).

Some of your measurements will be taken relative to a fixed point. Once you have chosen a Point 1, you can push the "set origin" button and Point 1 will be frozen until you push the same button (now labeled "clear origin") again. After you have chosen an origin, all measurements are taken relative to that point, even if you change images. If you get confused, or if there is a red line on the image that won't go away, just push the button once or twice until it is labeled "set origin" again and the boxes should empty and any red lines will go away.

It will be easiest to take the data if your screen resolution is 1024x768 or 800x600. This will make the images larger on the screen and it will be easier to point to individual pixels.

As you take data, remember that you do not have to do it all at once. It is a somewhat tedious process, and the more care you take the better your results will be. Don't be afraid to take the same data point repeatedly until you are happy with it, and take a break whenever you need to; your data quality will suffer greatly as you tire. You may write the data down, but it is easier to copy and paste it electronically into a text file or spreadsheet. Be sure to save your data regularly as a safeguard against computer malfunction!

1. Measure the radius of the puck, and the radius of the hole in the puck, using the image above.
2. What are some objects with the same mass as the puck? What is the density of the puck? It's original destiny was not to be a puck in a College Physics lab experiment; what do you think it is made of, and can you determine what it was originally intended for?
3. Using the Java program, set your origin at the center of the puck in the first image. Then push the back-frame ("|<") button to place the puck at the end of the run and choose your second point at the center of the puck. It will be more difficult to locate the exact center as the puck accelerates: the hole in the middle will appear elongated due to its motion while the image was captured.

   The distance between the two points is the total distance traveled by the puck, denoted "dl_total"; record this distance. The angle is the angle of the incline. Subtract this angle from 360 degrees; the result is the angle of elevation. Record both angles.

   How does this angle compare with the value you found empirically during the first class meeting? What part does the size and mass of the puck play in determining the optimal inclination angle for the rolling descent?

4. Using the Java program, set the origin on the 20 cm mark on the meter stick (corresponding to the left red dot below) and measure the distance in pixels to the 90 cm mark (corresponding to the right red dot below), making sure that the angle is the same as that recorded for the incline:
   

   You will be able to convert the x and y pixel coordinates to centimeters by multiplying each pixel value by 70 and dividing by the number of pixels between the 20 and 90 centimeter marks. Compute this conversion factor. Use it to convert dl_total to cm; use this value below.

5. Submit your values for the inner and outer radii of the puck, the angle of elevation and your conversion factor to me at the beginning of class the 2^nd week of the quarter.
6. Using the Java program, set your origin back to the center of the puck in the first image at the same coordinates as before. Take the x and y coordinates of the center of the puck for every image, attempting to keep the angle as close to the incline angle as possible for every data point. This will be harder for the first few images due to the limited resolution, but rapidly get easier as the puck accelerates.

   Make sure you have as many (x,y) coordinates as images.

7. Plot the x coordinate (in pixels) against the time (use t = 0 for the first image). Do the same for the y coordinate. Use a computer program (spreadsheet, Mathematica, etc.) to do all of your plotting. What is the shape of the plot? Describe it in mathematical and not visual terms.

   Data entry errors will manifest themselves as sharp discontinuities in the plot(s). If you have any, fix them.

8. Using the Java program, measure the angle between the "4 dot" line and the x axis for every image. This will get slightly harder as the puck accelerates. Begin by clearing the origin and moving the puck back to the first image. For each image, click first on the center of the puck, then on its edge along the 4 dots. Be sure to measure all of your angles in this way. The resulting red line should as nearly as possible bisect the white area on the image corresponding to the 4 dots.

   It is helpful to set the origin in the center of the puck for each of these measurements; that way you can be sure you have it in the same position you recorded earlier, and move the edge of the line easily to get the best bisection of the four dots. If you do this, you will have to clear the origin before advancing to the next image.

   Make sure you have as many angle data points as coordinate data points when you are finished.
9. Plot your angular data as a function of time. Note the cyclical nature of the plot as well as the fact that the cycles get shorter and shorter.

   Except for the cyclic nature of the angles, there should again be no sharp discontinuities. If there are any, they represent data entry errors; fix them.

10. Everyone's values for the puck radii, the angle of elevation, and the conversion factor should be the same. Because of errors in measurement, there will be small variations between some student's values. Since these values were physically the same for every data set, we will compute the mean and deviations for each value, and use those means for all subsequent calculations.

    Compute the mean (or average, denoted by the Greek letter "mu": m) for each value over all of the trials (data sets). We will denote these as "r", "r_in", "q" and "cm/pixel" (q is the Greek letter "theta").

11. Compute the absolute deviation of r, r_in and q over the set of trials.

    To find the absolute deviation of a data set, first find the absolute value of the the difference between every data point and the mean (its deviation from the mean). The absolute deviation is defined as the largest of these differences.

    Repeated trials of an experiment rarely turn out exactly the same, no matter how hard you try to re-create the initial conditions. This is a result of random (or statistical) error, which is complementary to systematic error. While systematic error is determined by the experimental design, random error is a result of living in the real world: there are so many factors affecting every trial that it is impossible to exactly duplicate any given trial. By performing repeated trials and measuring the absolute deviation, we can quantify the random error and ensure that we do not retain more significant digits than our experiment allows.

12. Compute the relative deviation of r, r_in and q over the set of trials.

    It is easier to evaluate the random error if it is expressed as a percent. This number is called the relative deviation, and is found by dividing the absolute deviation by the mean and multiplying by 100% (because "percent" literally means "over 100"). Note that the relative deviation is unitless.

    We sometimes use absolute or relative deviation to measure the discrepancy between the measured and advertised (or standard) values for some number. For instance, in the Winter Lab Project, we will compute the relative deviation between the measured and advertised values of several electrical components. In these cases, we are comparing the measured value to the advertised value, so the relative deviation is found by dividing by the advertised value. If we are comparing to some standard value (ie., 980 cm/s² for g), we find the relative deviation by dividing the absolute deviation by the standard or accepted value.

13. Compute the standard deviation of r, r_in and q over the set of trials.

    The absolute and relative deviations measure the variability of a data set, but not the "spread" (how clustered or spread out the data is). If we assume that the deviations in a set of repeated trials are due solely to random error, we can expect the data to follow a Gaussian or normal distribution: the familiar "bell curve", centered on the mean:

    

    In such a distribution, approximately 68.27%, or just over two thirds, of the data points will fall in the range described by the mean plus or minus one standard deviation (within the red lines above). The standard deviation is denoted by the Greek letter s ("sigma"), and is computed by squaring all of the absolute deviations, adding the squares, dividing by one less than the number of data points, and taking the square root:

    s = (S (x_i - m)² / (N-1))^1/2

    (S indicates summation.) s has the same units as the absolute deviation.

    Note that the standard deviation is a more reasonable measure of the variability of the data set.
14. Submit your means, standard deviations and all three of your data plots to me at the beginning of class the 3^rd week of the quarter.
15. As was previously mentioned, the NTSC video standard determined the temporal resolution (.0333667 seconds between frames). The standard also determines the spatial resolution by defining the number of lines in each frame to be 480. Since the aspect ratio (which defines the shape of the images) is 3:2, each line can be broken up into 720 pixels (6 black pixels were cropped from the left side of each frame to yield images which are 714 pixels wide, and the useful part of each image was contained in the 120 lines in each image). These limitations in temporal and spatial resolution are examples of systematic error.

    Convert the x and y coordinates to centimeters (cm).

    All coordinates and distances should be in cm from this point on.

    The raw data was accurate to a resolution of at most one pixel, depending on how careful you were using the program (above). What is the accuracy of the converted coordinates, in cm?
16. Use the Pythagorean Theorem to compute the distance traveled between each image. You will have one less distance data point than coordinate data points.

    If this bothers you, consider the fingers of one of your hands: supposing you have 5 fingers, you will only have 4 spaces between the finger tips.

    We will denote the sum of the distances as "dl_sum". Is this the same as dl_total? Why or why not?
17. The systematic error in our experiment means that there is uncertainty in our data. By saying that the raw data was accurate to a resolution of at most one pixel, we are saying that the uncertainty is at least one pixel. Further, whenever you perform computations with uncertain data, the uncertainty can increase; this is called propagation of uncertainty.

    Assuming that the uncertainty in our coordinates is 1 pixel, the absolute uncertainty in each coordinate, in cm, is equal to the accuracy you computed above. Let us call that value "Dx" ("Delta-x"). In order to numerically evaluate the uncertainty propagation in our computation of the total distance traveled by the puck, we have to compute the relative uncertainty for each value of x and y. We will call these "Dx_rel" and "Dy_rel". For each coordinate, Dx_rel = Dx / x (and the same for y). Compute the relative uncertainties in your x and y coordinates.

    Using the root mean square, or RMS method of propagating uncertainty, the uncertainty for the sum of the individual distances between images is the square root of the sum of the squares of the uncertainties for every x and y coordinate, divided by the square root of the number of data points:

    RMS = (S (Dx_rel² + Dy_rel²) / N)^1/2

    Compute the RMS uncertainty.

    You can compute the absolute uncertainty in dl_sum by multiplying the RMS uncertainty times dl_sum. This is the "expected uncertainty" due to propagation. Now compute the absolute difference between dl_sum and dl_total; this is the "actual uncertainty". If the expected uncertainty is greater than the actual uncertainty, dl_sum and dl_total can be considered consistent. Are they?

18. Submit your dl_sum value and the absolute uncertainties you just computed to me at the beginning of class the 4^th week of the quarter.
19. Compute the average velocity between each image. We will call the velocity between images i and j "V_i,j". Of course, you will have the same number of velocity data points as distance data points.
20. Since it is an average, we will assume that the puck had velocity V_i,j at the midpoint between images i and j. Plot the velocity against the time (your first point will be (.0333667*(1/2), V_1,2), your second will be (.0333667*(3/2), V_2,3), your third will be (.0333667*(5/2), V_3,4), etc.). Why does the plot look the way it does? (Remember that the puck wobbled as it traveled down the track, and possibly rubbed against the sides of the track as it did so.) What type of function would best fit the velocity plot? Why would you expect this?
21. If you have N images, you have computed N-1 velocities. The time between the first and last velocity points is then t_last - t_first = ((N-1)-1/2)*.0333667 - (1/2)*.0333667 = (N-2)*.0333667.

    If this bothers you, consider the five-fingered hand we considered above. It had only four spaces between fingertips, and it has only three distances between those four spaces. Those 3 distances correspond to the time between the first and last velocity points.

22. We usually assume that acceleration is constant. While this is almost certainly not true in this experiment, we need to find a number that represents the best fit of our data with a constant acceleration. Since we have a list of velocities as a function of time, we can find the straight line which best fits that data; its slope will be our average acceleration.

    In order to compute this slope, we are going to perform a least squares fit (also known as a linear regression).

    In general, for n pairs of data points (x_i, y_i) with mean (X, Y), the slope (m) and intercept (b) are given by

    m = (n * X * Y - S(x_i * y_i)) / (n * X² - S(x_i²))

    b = Y - m X

    where the S denotes a sum over all of the data pairs.

    Here,
    • n is the number of velocity data points you have;
    • x_i are the times midway between each image (x₁ = .0333667*(1/2), x₂ = .0333667*(3/2), etc.);
    • X is the average of all the x_i;
    • y_i are the velocity values (y₁ = V_1,2, y₂ = V_2,3, etc.); and
    • Y is the average of all the y_i.

    So if you had 10 velocities, the slope equation would be

    m = (10 * X * Y - (x₁ * y₁ + x₂ * y₂ + ... + x₁₀ * y₁₀)) / (10 * X² - (x₁² + x₂² + ... + x₁₀²))

    That slope is your average acceleration (denoted "A"). The y-intercept is the initial velocity (denoted "V₀").
23. Submit your average acceleration and your initial velocity to me at the beginning of class the 5^th week of the quarter. I will then disseminate the class' results to you.

    Compute the mean and standard deviation of the accelerations. What does this tell you about the experiment, and about your data set in particular?

24. The Earth is an oblate spheroid (a slightly squashed sphere), and the acceleration due to gravity (denoted "g") depends on the distance you are from the center of the Earth. This means that g depends on both your latitude and your altitude. You can compute g from Helmert's Equation:

    g = 980.616 - 2.5928 cos (2 f) + 0.0069 cos² (2 f) - 3.086 * 10^-6 H

    where f ("phi") is your latitude and H is your altitude in centimeters above sea level. Assume that Blue Ash, Ohio is at 39.25 degrees North Latitude and is 850 feet above sea level. Use these values to compute g.

    Note that if you use Excel to compute sines and cosines, it expects you to convert degrees into radians before taking the sine or cosine.

    Everyone should get the same value for g. Make sure your value is correct before you proceed.
25. As you will learn from your text, an object moving down an incline feels an acceleration parallel to the incline of

    A_P = g sin q,

    where q ("theta") is the angle of the incline. The component of its acceleration normal (perpendicular) to the incline is

    A_N = g cos q,

    Compute A_P and A_N. You should expect A_P to be much smaller then A_N because q is so small.
26. Since the angle of the incline was the same for every trial, the acceleration due to friction between the puck and the sides of the track is

    A_f = A_P - A.

    Compute A_f.
27. You will also learn from your text (and the following section) that force is defined as the product of mass and acceleration, momentum is defined as the product of mass and velocity, and the force is equal to the change in momentum over time. Solve these equations for momentum and use A to compute the change in momentum (denoted "Dp").

    Use A_f to compute the force due to friction (denoted "F_f"), and A_N to compute the force exerted normal to the incline by the weight of the puck (denoted "F_N"). Since we have measured distance in centimeters, mass in grams and time in seconds, we are using the CGS system of units. The CGS unit of force is the dyne.

28. Your text will also teach you that the puck loses potential energy (denoted "U") and gains kinetic energy (denoted "K") as it rolls down the incline. The change in potential energy (denoted "DU") is the product of the mass, the acceleration due to gravity and the change in the y coordinate (taken as a positive distance). The change in kinetic energy (denoted "DK") is the difference between the squares of the final and initial velocities, multiplied by half of the mass.

    Since the velocities fluctuated significantly, compute your initial and final velocities from your linear regression equation. Use t = 0 and t = (N-1) * .0333667 seconds for this purpose.

    Compute DU and DK. Since we are using the CGS system in this experiment, these numbers will be in units of ergs. The erg is a very small unit, so these numbers will be relatively large compared to the others you have been computing.

29. Submit your DU and DK values to me at the beginning of class the 6^th week of the quarter.
30. Up until now, we have dealt only with the translational motion of the center of mass of the puck as it rolled down the track. We will now analyze the rotational motion of the puck. When you took your angular data from the images, you essentially treated the center of the puck as the origin for each value of the angle. This was equivalent to adopting a coordinate system which moved with the center of mass. This is an example of the principle that for any mechanical motion, you can treat the translational and rotational degrees of freedom independently.

    While the applet provided the angles in familiar degrees, the standard measure for angles is the radian. There are p radians in 180 degrees. Convert all of your angles to radians.

31. For every value greater than p, subtract it from 2 p and plot the data again. What mathematical function does this represent?

    The period is the time for one cycle; what do the decreasing periods indicate about the angular velocity? (See your text.)

32. While both of these plots are informative, neither is useful for our immediate purposes. Each time the angle rotated past 0 radians, it started over again near 2 p radians; we need the angles to be a monotonically decreasing function of time. The following procedure will give us such a function.

    Starting with the data from the first angle plot (which we will call the "original" data), adjust it as follows:

    1. When the original angles start over near 2 p, begin subtracting 2 p from each angle until...
    2. when the original angles start over near 2 p again, begin subtracting 4 p from each angle until...
    3. when the original angles start over near 2 p once again, begin subtracting 6 p from each angle until... And so on.

    This process will adjust your data as follows: the red data points marked "A" are unchanged. The black data points marked "B" are translated down by subtracting 2 p so that they continue where the "A" points left off. The "C" points are translated down to where the "B" points left off by subtracting 4 p from them, etc.

    

    The plot from step 9 should resemble the black data points, while the plot you are about to make should resemble the red ones.

    Plot the adjusted, monotonically decreasing angular data. This is the form of the data we will use in the rest of the analysis.
33. Compute the total angular displacement over the trial (denoted "dq"). We will interpret dq as an "angular distance", and therefore it is positive. As a check on your accuracy in taking data, compute the total arc length (denoted "ds") traveled by the puck by mulitplying its radius by dq. Compute the relative deviation between ds and dl_total as

    100% * | ds - dl_total | / dl_total

    How careful were you?
34. Compute the average angular velocity between each image. We will call the angular velocity between images i and j "w_i,j" ("omega"). As before, you will have one less angular velocity data point than angle data points.

    Note that this step and the next two are strictly analogous to steps 19 through 22 for the linear distance data, except that the angular velocities and acceleration are negative, due to the clockwise rotation of the puck.

35. As before, assume that the puck had angular velocity w_i,j at time (i + j) / 2. Plot the angular velocity against the time. Why does the plot look the way it does? What type of function would be the best fit the angular velocity plot? Why would you expect this?
36. Using linear regression on the angular velocity data, compute the average angular acceleration over the course of the entire data set as the slope of the best fit line.
37. Submit your relative deviation for ds, average angular acceleration and initial angular velocity to me at the beginning of class the 7^th week of the quarter. I will then disseminate the class' results to you.

    Compute the mean and standard deviation of the angular accelerations. What does this tell you about the experiment, and about your data set in particular?

38. The moment of inertia (denoted "I") is the rotational analog to mass. For a disc with a hole in it of radius r_in,

    I = m (r² + r_in²) / 2.

    Compute the moment of inertia for the puck. Everyone should have the same value for I. Be sure that you do before you proceed.
39. As described in your text, torque (denoted t, pronounced "tau") is the rotational analog to force, and is equal to the product of the moment of inertia and the angular acceleration. Compute the average net torque. Note that t is negative; this too indicates that the net motion of the puck was a clockwise rotation.
40. Your text also defines the rotational kinetic energy and angular momentum in analogy to the translational case. In the same fashion as above, compute the your initial and final angular velocities from your regression equation, the change in rotational kinetic energy (denoted "DK_rot") and the change in angular momentum (denoted "DL") between the first and last images of your data set.
41. Now we are ready to tie together our analyses of the translational and the rotational motions of the puck.

    Conservation of Energy tells us that for the puck, the change in potential energy is equal to the change in translational kinetic energy plus the change in rotational kinetic energy plus the energy dissipated by friction (denoted "E_f"), or

    DU = DK + DK_rot + E_f.

    Compute E_f.
42. The rate of the change in energy with respect to time is the power ("P"). Compute the average power lost due to friction. Compare this to the power required by a 60 Watt incandescent bulb. (Be careful with units here!)
43. The efficiency of any process can be expressed as the ratio of the useful energy gained to the total energy applied, or alternatively, as one minus the ratio of the wasted energy to the total applied energy. Since the potential energy is applied by gravity and assuming that the change in total kinetic energy is the desired outcome of the experiment, compute the efficiency of the experiment. Express it as a percentage. What are the units of efficiency? (Remember, "percent" is not a unit!)
44. If we assume that the frictional force F_f was constant, we can compute the work done by friction as

    W_f = F_f dl_total.

    Part of this work was lost as E_f and the remainder guaranteed that the puck did not slip as it rolled down the incline. Since we have assumed the puck did not slip as it rolled, the friction between the edge of the puck and the floor of the track must have contributed a torque of

    t_s = m_s F_N r,

    where m_s ("mu") is the coefficient of static friction between the puck and the floor of the track (why static and not kinetic?). What is the direction of rotation associated with t_s?

    Using the fact that

    W_f - E_f = t_s dq,

    compute m_s. Is this a reasonable value? Was it necessary to know the mass of the puck in order to calculate m_s?
45. Submit your W_f, E_f and m_s values to me at the beginning of class the 8^th week of the quarter.
46. Compute the mean and standard deviation of the class results for m_s.
47. Describe our analysis conceptually using the terms acceleration, energy, force, friction, torque and work. Identify the assumptions we made and evaluate how changes in them would affect our analysis.
48. Which, if any, of the error analyses we did could be used to evaluate the precision with which you took your data?
49. How could we reduce the effects of friction in this experiment? Are there frictional effects that we do not want to reduce?

• J. J. Aubert et al., Experimental Observation of a Heavy Particle J, Physical Review Letters, 33, 1404-1406 (1974)

  This paper and the next describe the first observations of a particle predicted by the fledgling Standard Model of Particle Physics. Since it was discovered essentially simultaneously by two different groups in two different laboratories, the particle has become known as the "J/y" after the names given by the two groups. Note the dates on which each paper was received by the journal. As you read the second paper, think about how these two papers are alike and how they are different, even though they describe the same result, found in two very similar experiments.

  In particle physics, p stands for proton, e^- for electron, e⁺ for a positron (identical to an electron except with positive charge) and m stands for a muon (a heavier brother to the electron). eV is defined in your text; the prefix "G" stands for 1 billion. Look up the terms "Cherenkov", "hadron" and "synchrotron".

• J. -E. Augustin et al., Discovery of a Narrow Resonance in e⁺ e^- Annihilation, Physical Review Letters, 33, 1406-1408 (1974)

  This paper reports on observations made at SLAC: the Stanford Linear Accelerator Center.

  A cross section describes collisions between particles. Resonance is another term for a peak in a cross section. Bhabha scattering is the process when an electron and a positron collide but yield only another electron and positron.

• K. Hirata et al., Observation of a Neutrino Burst from the Supernova SN1987A, Physical Review Letters, 58, 1490-1493 (1987)

  The Kamiokande detector was originally built in a mine in Japan to search for proton decay. The detection of neutrinos (very light neutral elementary particles denoted by the Greek letter n, pronounced "nu") from the first observed supernova of 1987 not only helped to verify the standard model of supernova physics, but also ushered in a new era of neutrino physics in which underground detectors are used to detect neutrinos from astrophysical sources.

  "g" is the Greek letter gamma, here used to denote high-energy photons released from radioactive decay. "PMT" stands for photomultiplier tube; look this one up. Compton scattering is the process when an electron scatters off of a photon. A Poisson distribution is a statistical distribution (like the Gaussian one discussed above) which describes the arrival of events.

  You can figure out what flux is by examining the units. What and where is the Large Magellanic Cloud?

• G. F. Smoot et al., Structure in the COBE Differential Microwave Radiometer First-Year Maps, The Astrophysical Journal, 396, L1-L5 (1992)

  COBE, the Cosmic Background Explorer satellite, was built to observe one thing: variations in the Cosmic Microwave Background (CMB) radiation. CMB radiation is essentially the photons left after the Big Bang when the universe cooled sufficiently to become transparent to photons. When was this, and why is its temperature now approximately 2.7 degrees Kelvin?

  Those variations are thought to be the seeds of all the structural elements in the observed universe, including you and me. Look up the terms "anisotropy", "correlation", "dipole" and "quadrupole". The images at the end of the paper, particularly the last one, are precursors to an image which may be familiar to you from NASA press releases.

©2007, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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