Otterbein University Department of PhysicsPhysics Laboratory 1500-1

Name: ______Partner’s Name: ______

EXPERIMENT 1500-1

INTRODUCTION TO PHYSICS LABS: GRAPHING & STATISTICS

Part I. INTRODUCTORY REMARKS

In a typical lab session, you will follow the handout provided to you and fill in your answers, measurements, etc. as you go. Be sure to show your work for all calculations. You will sometimes be asked to make extra plots or print something, which you will staple to the back of the provided worksheet. Be sure everything you hand in clearly bears your name and the lab number, and is stapled together. Remember that you will be marked on correctness, completeness, and important details like units, labels on graphs, and significant figures. When drawing graphs you must use a ruler.

What follows is a short compilation of basic terms and concepts which are important in every lab setting. Please read through the material carefully.

  1. Units.

In 1999, NASA lost a $125 million spacecraft called the Mars Climate Orbiter after it improperly entered the Martian atmosphere and disintegrated.[1] NASA performed part of the work on the spacecraft themselves and contracted other parts of the work to a third-party. NASA was working in SI units (meters, kg, etc), but the contractor was working in English units (feet, pounds, etc.). The lack of communication about which unit system each party was using and the failure to properly convert between units resulted in an error which led to the loss of the craft. You should learn from this mistake and always record the units of every measurement that you take and convert between units when necessary.

B. Significant Figures.

  1. Numbers should be given with the number of digits required to express the precision of the measurement. For instance, a value 2.1 implies an uncertainty of ±0.1; a value 2.100 implies an uncertainty of ±0.001; a value of 2.1×102 implies an uncertainty of ±10.
  1. When two numbers are multiplied or divided, the number of significant figures is the same as the number of significant figures of the least precise factor. Thus 1.134 × 2.1 is 2.4 to within the implied precision of the original numbers. To record 1.134 × 2.1 = 2.3814 is misleading, since your measurements aren’t accurate enough to determine the last three digits. The extra digits are no more meaningful than if you made them up.
  1. When two numbers are added or subtracted, the precision of the result is in the last decimal place of the least precise term. Thus 1.134 + 2.1 is 3.2 to within the implied precision of the original numbers.
  1. It is possible to lose precision by rounding at intermediate steps in a calculation. In this day of calculators and computers, you might as well play it safe by keeping all the digits your calculator will hold, and rounding off to the right number of significant figures only at the end.

Question: Display the answers to the following problems with the correct units and the correct number of significant digits:

C. Percentages

When two numbers disagree (for instance, the results of two different experiments, or an experimental value and a theoretical value), it is often useful to express the difference as a percentage. This can help you judge whether the discrepancy is large or small. Large or small compared to what? One useful comparison is with one of the values. The percent difference of B from A is defined as:

Here, you’re comparing the difference, B−A, to some reference value, A. For instance, A might be a theoretical value of some quantity you’ve just measured, and B your experimentally measured value. The same formula expresses the percent change, if A represents the initial value of a variable and B represents the value after some change.

Question: If you are going 85 mph and the speed limit is 70 mph, by what percent are you exceeding the speed limit?

D. Uncertainty

In any experiment, there are variables you cannot control that may affect the result of your measurement. For instance, if you are driving down a bumpy road, the needle of the gas gauge on your car may jump up and down, giving you different readings each time you look. The accuracy of your measurement is also limited by your measuring instrument(s): for example, with an ordinary meter stick, you probably cannot measure lengths more accurately than a millimeter or so. The effect of such factors on the result of your measurement is called uncertainty.

Uncertainty comes in because of two kinds of imperfections in real experiments: random errors, and systematic errors.

Random errors give readings that are as often high as low. The bouncing needle on a car’s gas gauge might be an example of a random error. This means you are uncertain as to the true gas level. This random uncertainty can be reduced repeating an experiment many times and averaging the results. You will be more confident (have lower random uncertainty) in your mean value than in the value of any single measurement.

Systematic errors, on the other hand, cause your measurement to be always too high, or always too low. An example would be a thermometer that is mis-calibrated, so that it always reads 2 ºC too high. If you know you have a systematic error, you can correct for it. For instance, if you know your thermometer reads 2 ºC high, you can subtract 2 ºC from all your readings. The only way to know if you have a systematic error is to test your instruments by measuring a quantity whose value is already known. It’s never possible to check against perfect equipment (which doesn’t exist) and so we always have a systematic uncertainty because we have to estimate the accuracy of our equipment.

By the way: don’t ever use the term “human error”: it means that someone made a mistake. If you made a mistake, fix it! Using this term is admitting that you should be flunked for not getting it right.
Question: Most materials expand as they get hotter, including rulers. On a hot day, does this introduce a random uncertainty or a systematic uncertainty to your measurements? Explain.

Question: Would switching from a ruler with markings every mm to a ruler with markings every cm increase the random or systematic uncertainty of your measurements of length?

PART II. GRAPHING & LINEAR REGRESSION

We often measure a quantity as a function of a (independent) variable. Namely, we allow one quantity to vary and take many measurements of a second quantity that depends on it. Examples are a position as a function of time, or the temperature of a steel rod as it varies along the length of the rod. Typically one collects the data (pairs of numbers) in a table. This data can be used to make a plot which enables us to discuss the dependence of the dependent variable (the measured quantity) on the independent variable (the parameter allowed to vary). We always plot the dependent variable versus the independent variable, i.e. the dependent variable on the vertical axis. Note that values in physics have units, which must be displayed in both tables and plots.

A. Plotting Data

  1. Never connect the dots when you plot your experimental results. You do not actually know the shape of the curve between your data points, since you haven’t measured it. Furthermore, if your data has any scatter in it, connecting the dots will exaggerate it. For instance, the graph below shows a more-or-less linear relationship between y and x, with some scatter; but you wouldn’t think so, from the graph on the right.

  1. Scales should be easy to read even at intermediate points. Scales such as 3 squares to represent 4 grams are not acceptable. A readable half-scale graph is far better than an unreadable full-page graph. It should be possible to choose a scale that is readable and spreads the data points over a large fraction of the provided grid.
  1. Coordinate axes (both x and y) must be labeled with a title and units as shown here.


  1. If possible, avoid the use of powers of ten; instead, use the appropriate prefixes of the Système Internationale (i.e. don’t use 3x10-3 m, use 3mm). If powers of ten cannot be avoided, the American Physical Society recommends[2] that it be associated with the unit label, thus:


  1. The origin should be included on the graph unless there is an excellent reason to do otherwise. Graphs with suppressed origins are usually misleading and may conceal important information such as intercepts.
  1. Hand-drawn graphs should be made on ruled graph paper. You will usually have graph paper included in the lab worksheet. If not, ruled paper is available from your instructor upon request.

B. Linear Plots

In physics, the relationship between two quantities is often linear (y=mx+b), and the slope, m, and y-intercept, b, are often quantities of interest that can be extracted from your data. To determine a slope or an intercept from a hand-drawn graph, draw a line that best fits the plotted data and measure its slope. (“Best fits” means that the line comes closest to passing through all of the points.) Do not calculate the slope from just two data points (unless specifically directed to do so); if you do, you will lose the benefit of using many data points to average out random fluctuations in your data. The slope intervals used should be drawn on the graph, and encompass all of the data that shows a linear relationship. If you use shorter intervals, the uncertainty in your slope will be larger. Show the calculation of the slope on your paper, so the intervals can be checked directly.

(Note that if you want to easily determine the intercept b your horizontal axis must begin at zero.)

ACTIVITIES:

  1. Find the slopes and the intercepts of the graphs displayed below. Be careful to quote results with appropriate units.

slope = ______

y-intercept = ______

slope = ______

y-intercept = ______

  1. Make a table and plot of the following function: d(t) = (17m/s) t + 5.5 m, where t is measured in seconds. Plot values between -1s and +5s at 1s intervals.

t (s) / d (m)

Is thisa linear relationship?

  1. The following data are given in the form of a table:


Plot the data and estimate the slope and y-intercept of the graph by “trying out” different possible slopes. Plot the best fit slope on your paper. Estimate an uncertainty for both quantities and list them below. (Hint: make a reasonable estimate for how much these values could vary and still give an acceptable fit.)

slope=______± ______

y-intercept =______± ______

  1. Now we will use some real data. Set the track up as advised by the instructor and make sure it is level in the “width” dimension. Measure the position of the cart on the track as a function of time. One person counts out the seconds with the help of a stopwatch, the others jot down the position of the front of the cart on a piece of paper as accurately as possible. Make a table and plot your results.

Is this a linear relationship? Should it be? Why or why not? (Hint: we will soon learn that a position vs. time graph is linear if there is no net force on acting on an object)

PART III. STATISTICS

i / Xi / di / di2
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Most experimental measurements are repeated several times, and it is very likely that the values obtained will differ from measurement to measurement. (Why?) For a set of measurements that are all equally trustworthy, it can be shown that the best estimate of the true value is the average or mean value. The meanof a set of N measurements X1, X2, … is written <X> or , and is defined to be

  1. In the above table, record the set of numbers 5.32, 6.42, 6.00, 5.85, and 6.23 in the vertical column. Calculate their mean (show your work):

  1. Having obtained a set of measurements and determined the mean value, it is helpful to state how much the individual measurements differ from the mean. The deviationdi of any measurement Xi from the mean value <X> of a set of such measurements is

As defined, the deviation may be positive or negative since some measurements are larger than the mean and some are smaller. The mean of the deviations is always zero.

In a second column of the data table, record the deviations of the numbers in the first column and verify by direct calculation that the mean of the deviations is zero.

  1. Since the deviation may be either a positive or negative number, it is convenient to use the square of the deviation as a measure of the scatter or dispersion of the measurement about the mean (squared numbers are always positive). The variances2 of a set of N measurements is defined to be:

The square root of the variance, s, is called the standard deviation or the root-mean-square deviation. Note that the standard deviation has the same units as the measurements X and is always positive; it is used to describe the random uncertainty of a set of measurements.

Find the standard deviation on these measurements. In a third column of your data table, record the square of the deviations from column 2. Calculate the standard deviation and list it below.

The uncertainty on any individual measurement is s, therefore each measurement would be quoted as X ± s. So, if you had only taken the first measurement above, s would be your error on that single measurement.

The uncertainty on your mean value gets smaller the more measurements you make and is equal to

Remember that the random uncertainty gets smaller as you take more measurements Therefore your measurement of the mean value would quoted as .

Quote the mean value with the uncertainty on the mean.

If your scientific calculator has the capability of evaluating the mean and standard deviation of a set of numbers, compare your results of a direct calculation with the results using your calculator algorithm. If you find they are different, check your calculator manual to see if it is due to the definition o fX> and s used by your calculator. Some use a formula for the standard deviation that has N in place of N–1. (The TI-36, for example, offers you a choice of buttons labelled xn and xn–1.) The formula with N–1 is the appropriate one in this course.

  1. Now, in order to have a real experimental data set to practice on, measure the time it takes the cart to go down the track five times. Prepare a data table to record the measurements, and calculate the mean and standard deviation of your measurements.

i / ti(s) / di (s) / di2 (s2)
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Show your calculations here. Also quote the mean along with the uncertainty on the mean.

What do you think is the main cause of random variations?

Estimate the random uncertainty on this measurement.

What would you guess is the biggest cause of systematic uncertainty?

Estimate the systematic uncertainty on this measurement.

Which is bigger (and therefore more important)?

Equipment:

-FCI + SCANTRON

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-Tracks, with book under one angle

-Carts (old ones)

-Stopwatches

-Rulers

-colored cups

Notes for next time:

-Explicitly walk through uncertainty on slope and intercept.

-Put more scatter on artificial ‘best fit line’ points

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