Physics 15 Lab Manual
Measurement, Uncertainty, and Moon Craters
Page 1
Measurement, Uncertainty, and Moon Craters
Please note: the introductory section for this lab is much longer than will be usual. Please
read it carefully. It contains ideas you will need throughout the course.
I.
Introduction to Measurement and Uncertainty
Physics is an experimental science. We talk about the “laws” of nature, and it may appear
to you that these laws are engraved in stone, or printed in some master textbook in the
Library of Congress, and that all you need to do is learn them.
Not true.
The foundations of this course are Newton’s “Laws.” That sounds rock solid until you
realize that we cannot prove Newton’s laws to be true! So where did these so-called laws
come from? Experiments.
Newton (and Galileo before him) spent some time thinking about the results of some actual
experiments and developed his “laws” as a set of hypothetical rules that seemed to describe
all the data. That does not constitute a proof of the laws. All we can say is that in many
thousands of experiments, no one has ever found a violation of Newton’s laws.
All of this is a long way of introducing the idea that experiments—not textbooks, problem
sets, and exams—are the real heart of physics. And the real heart of any experiment is
the data. So you can see how it is very important to understand how to think about data:
how to understand what it shows you, and what it does not show you.
An Unfortunate Truth: The analysis and treatment of data can require sophistication,
wisdom, incredibly ugly mathematics, and the kind of judgement that only comes from
years and years of practice.
A Not-So-Unfortunate Truth: many experiments can be analyzed using straightforward
and simple techniques that are easy to understand. The purpose of this lab is to introduce
you to a few of these techniques, and to explore the physics of crater formation.
II.
Types of Error
Nothing about the physical world is known exactly. All our knowledge of this universe
comes from measurements, and we simply cannot measure perfectly. For example, in
this course you will encounter the gravitational constant G. We currently think that this
constant has the value (with our standard system of units) 6.67259 × 10−11 m3 kg−1 sec−1 .
On the other hand, our uncertainty in this value, which comes from the “errors” in our
measurements, is ±0.00085 × 10−11 m3 kg−1 sec−1 . A more useful way1 of thinking about
1
Here is how to write the value of a measurement together with its uncertainty: G =
(6.67259 ± 0.00085) × 10−11 m3 kg−1 sec−1 .
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Measurement, Uncertainty, and Moon Craters
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this uncertainty is to express it as a percentage of G. Our uncertainty in the value of G is
about .01%, or “one part in 104 .” This is actually quite large, considering the sophistication
and effort that physicists have put into measuring G in the 300 years or so that we have
been working on it. It is equivalent to a 6 inch uncertainty in the length of a mile, and
any skilled surveyor could do better than that. In fact, G is the most poorly known of all
the fundamental physical constants.
This uncertainty in our knowledge of G comes from two sources: random errors and
systematic errors.
Error 6= Mistake
In the context of science, the word “error” does not mean mistake or blunder. It is not
that we don’t all make mistakes, but we (hopefully) can catch and correct them. The
essential point is that even after we have corrected all our mistakes, there are still limits
to how well we can know something. The purpose of this lab is to give you a feel for what
those limits are, and more important, how to quantify them.
Random Errors
These occur because there are a large number of unpredictable things that can happen
during an experiment. The power line voltage could fluctuate, or the floor could vibrate,
either one of which could affect the reading of a delicate instrument. Or, when trying to
estimate tenths of a millimeter while reading from a meter stick, we might make small
errors in judgement. Or, the object whose length we are measuring could have an uneven
end. The important point about all of these errors is that they are random—which lets us
apply some statistical tools. We can’t make random errors go away, but we can develop
ideas about what is a reasonable variation in our measurements, and what indicates a
problem.
Systematic Errors
This type of error is a real problem. For example, someone could give you a meter stick
that was printed incorrectly, so all the divisions were only 90% of “true” size. Then, if you
use this meter stick to obtain data on a falling object, the meter stick will incorporate a
systematic error into your calculation of the acceleration due to gravity, g. The difficulty
here is that unless you think to check the calibration of your meter stick, there is no way
to tell that there is an error! Especially if you are the first person to measure g, so there
is no “standard answer” to compare to.
Here is another systematic error that you are much more likely to encounter: suppose,
when reading with the meter stick, you don’t always look at it straight on. Well, if you do
this first to one side, and then to the other, and by slightly different amounts, this error
will act like a random error. But suppose, instead, that you always hold your head to
the same side, and always by about the same amount. You would then be introducing a
systematic error—all your length measurements would be incorrect in the same direction
and by about the same amount. (By the way, this particular problem is known as parallax
error.)
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Measurement, Uncertainty, and Moon Craters
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Unfortunately, systematic errors can be tough to spot, and there is no one presciption
for dealing with them. But you should be aware, and always asking yourself if your
measurement technique could be introducing systematic errors.
III.
Uncertainty
The combined effect of all these errors is to make us “uncertain” about things we measure.
But please! The word “uncertainty” has a VERY specific meaning in physics. You can be
uncertain about how well you did on a test, but that is not the same kind of “uncertainty”
as you will encounter in lab.
For example, suppose you are using a meter stick to measure the length of something like
the bolt in the figure below. The smallest divisions on the meter stick are .1 cm (1 mm)
apart. The end of the bolt clearly lies between 3.3 cm and 3.4 cm, but is not precisely on
either one. You could quite reasonably decide the uncertainty in the length is 1 mm, and
since the end is closer to 3.4 cm, you would record the length as 3.4 ± .1 cm.
Figure 1
It is worth pointing out that you could probably do better. You can easily tell which half
of the division the end lies in, so a less uncertain (but still reasonable) value for the length
is 3.40 ± .05 cm. You could also interpolate between the actual marks on the ruler, and
decide, for example, that the length of the bolt is 3.37 ± .02 cm. Interpolation is something
that you can get better at with practice. On the other hand, it is a good idea not to get
too carried away. . .
Here is another example. Suppose you use a balance to weigh an object to determine its
mass, and the balance has divisions of .1 kg. You weigh the object and find its mass to be
20.0 kg. Then your lab partner weighs it and finds 20.3 kg. You weigh it again and find
20.1 kg. Clearly we now have to think—what should you write down for the mass of the
object? It should be obvious to you at this point that whatever you do write down can’t
possibly be “exact.” So you will also have to determine an uncertainty and record that
too.
It turns out (and this is proably not surprising) that the best estimate for the mass of
this object is found by taking the average of the three measurements. Any calculator will
dutifully tell you that the average is “exactly” 20.133333333333333333333333333. . .
There is a problem with this result! To write it with complete accuracy would require an
infinite number of digits—but more to the point, more digits than our measuring device
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is even capable of producing. That means that most of the threes in the answer are
completely meaningless!! We can determine which of the digits have meaning, and which
are junk, by estimating the uncertainty in the result we get by taking the average. How do
we do that? Look at the original data. All the data points are (roughly) within 0.2 kg of
the average. So the uncertainty can’t be zero, but it can’t be huge either—an uncertainty
of 0.2 kg seems about right. (In many instances in this course you will be estimating your
uncertainties by this sort of “eyeball” method. In just a bit, we will develop a more precise
way of doing this, but it requires more data points.)
Now let’s condsider the average again, to eliminate those meaninless digits. We are about
to encounter the dreaded problem of significant figures.
There are lots of ways to make significant figures complicated, but let’s try for a way to
make them simple:
Any measured quantity cannot be written with
more precision than its uncertainty.
In our example the uncertainty is 0.2 kg, so we round our result to have the same number
of digits after the decimal place as the uncertainty, which gives us a best estimate and
uncertainty of 20.1 ± .2 kg. Don’t make the mistake of thinking you have to round to the
nearest .2 kg. 20.1 is clearly a better estimate than 20.2!
In the above example, the estimate for the mass has three significant figures. The uncertainty has only one significant figure, and appears in the same decimal position as the
least significant figure in the measurement. As a general rule of thumb, you should always
express your uncertainty to only one significant figure. It is the uncertainty! What kind
of precision can you really claim for it?
If you are unsure about how many significant figures a number has, use scientific notation
so that reading from left to right the first non-zero digit is just to the right of the decimal
point. In our weighing example, that would be 0.201×102 kg. Now express the uncertainty
to the same power of 10: 0.002 × 102 kg. The decimal position of the uncertainty (third)
tells you how many significant figures you have in your measurement (three).
One final point: When measuring something, pay careful attention to the number of significant digits and the uncertainty. BUT, when calculating a result from those measurements,
don’t worry about significant figures during the intermediate steps of the calculation. Let
your calculator keep track of all those extra digits until it is time for you to write down
the final result and its uncertainty. Only then should you work out the correct number of
significant figures. That way you will avoid the accumulation of rounding errors.
IV.
How many is enough?
If I ask you to repeat a measurement enough times so that you are sure of the answer, you
are likely to ask (in a tone that some might interpret as whining) “how many is enough?”
Whining aside, that is an excellent question. We will try to answer it with a simple
discussion. It is possible to actually derive the answer to this question, for which you
should see the books on error analysis on reserve for the course.
Physics 15 Lab Manual
A.
Measurement, Uncertainty, and Moon Craters
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The Mean
Just to have something concrete to talk about, let’s continue with the weighing example we
used before, but with a very precise balance. (How precise is unimportant. Imagine that
it has divisions of .001 kg.) The first thing to note is that this object has a fixed number
of atoms in it, and so has a “true” weight, Wtrue . Alas, this true weight is unknowable.
The best we can hope for is to come up with a really good estimate of Wtrue , which we
will call West , or W for short.
Suppose that we were very patient, and could weigh our object an infinite number of times.
Because of random errors we will not get the same answer every time. But if there are
no systematic errors then there will be no bias towards higher or lower weight. In other
words, on average we will record a weight above Wtrue just as many times as we record
one below. And, on average, the amount by which we miss will be the same in both
directions. Our range of recorded weights will be symmetric about Wtrue . Looked at from
this perspective, it should be fairly obvious that the best estimate of Wtrue is the average
of all our measurements, often called the mean.
Of course, you are not that patient. You won’t take an infinite number of measurements.
But it still turns out to be the case that your best estimate for Wtrue will come from taking
the average of your sample of measurements.
B.
The Standard Deviation
The next thing to consider is just how our measurements are distributed around Wtrue ,
aside from symmetrically. If we are reasonably careful, there probably aren’t any measurements that are way far away from Wtrue . On the other hand, there are probably lots
of measurements that are really close to Wtrue . And, as we look at weights that are farther
and farther to either side of Wtrue , there are probably fewer and fewer measurements, with
the number falling smoothly to zero as we eventually consider the way far away values. In
fact, if we made a graph that had the number of measurements on the vertical axis, and
the recorded weight on the horizontal axis, it would have the shape you probably know as
a bell curve, although it is more properly known as a Gaussian distribution.
Figure 2
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It would probably not surprise you to know that the width of our distribution of measurements (how fat the bell is) is related to the uncertainty in our measurement of W . There
are actually a number of ways to express the uncertainty in a result. We will use one
common one, but you should keep the idea somewhere in the back of your mind that there
are alternative definitions that are commonly used.
Since the Gaussian curve depicted in Fig. 2 is a well defined mathematical function (please
look at the references), it has well defined mathematical properties. One of these is known
as the standard deviation, often written σ (the Greek letter “sigma”). σ has the property
that when we take an infinite number of measurements, 68% of those measurements will
be within ±σ of Wtrue . But, of course, you are not going to take an infinite number of
measurements! So the best you can do is estimate the standard deviation. And, as it turns
out, just as the best estimate of Wtrue is the mean of your measurements, the best estimate
of σtrue is the standard deviation of your measurements, which you calculate according to
the following recipe:
Suppose you have N data points. Let’s name them x1 , x2 , x3 , . . . xi , . . . xN . Then the
standard deviation σ is defined to be
v
u
N
u1 X
t
(xi − x̄)2
(1)
σ=
N i=1
where x̄ is the average of the N data points.
In words: Take the difference between each data point and the average, and square
it. Add up all these squared numbers and divide the total by the total number of
data points (notice that we are computing an average of these squared differences).
Finally, take the square root.
C.
The Uncertainty
Here is the last part, but it’s subtle. It is also important.
What is the relation between σ and the number of times you repeat the measurement?
Notice that σ does not get smaller as you take more data! In fact, we defined σ for the
case that we took an infinite number of points! Does that mean we are stuck, that no
matter how hard we try, we can’t reduce the uncertainty in our measurement below the
standard deviation? Not at all. If we really did take a huge number of measurements,
then we would really nail down the exact shape of that bell curve. If we did that, then we
would have a very precise idea of where Wtrue is—right at the peak in the curve. And our
uncertainty would be very small.
This then is the essential idea of repeated measurements: the more data you take, the
better you know the shape of that bell curve. The better you know the bell curve, the
better you know where its peak is and the smaller your uncertainty.
All of that improved knowledge is represented in the following formula:
σ
σm = √
N
(2)
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Here, σ is the standard deviation of your sample of data, calculated according to Eq. 1,
and N is the number of data points in your sample. σm is your uncertainty in W , which
you found from taking the mean (average) of your sample of data. It has an official name,
the “standard deviation of the mean,” which is so obscure as to tell you nothing. It is
sometimes also called the standard error , which is a bit more meaningful.
You should look at Eq. 2 for a bit. You should be able to see that taking more data points
improves things two ways: your sample σ becomes a better estimate of σtrue , and the
increased value of N in the denominator makes the uncertainty smaller.
D.
OK, so how much is enough?
√
Well, it turns out that making things better by N is pretty tedious. You stay up all
night and take 100 times more data points, and only improve your uncertainty by a factor
of 10. That’s a pretty lousy return on your investment.
So, let’s pay attention to the standard deviation. You have taken enough data when
your estimate of σ is good enough. For practical purposes, we’ll say your estimate of σ
should lie with about 25% of σtrue . You can improve things by taking more data, but that
improvement will be slow. Of course, you don’t know σtrue . So a better idea is to start
taking measurements, and once you have at least 3, start re-calculating the mean and σ
after every point. (Most scientific calculators can do this for you. Learn how to use the
fancy functions on the one you own.) When your succesive values of σ are all within about
25% of each other, you have taken enough data. You might take one more measurement
just to be sure. . .
Please note that you have not bee given a rigid rule! There may be circumstances where this
“rule of thumb” isn’t good enough, and you require a very large number of measurements.
Or there may be circumstances where you just need a rough number. And sometimes, a
single (careful) measurement will give you all the precision you need. The important point
is that judgement is required. With any luck, this lab is a first step toward developing
that judgement.
E.
The Cruel Part
It is possible to show mathematically that if you take X number of measurements, your
measurements of σ should be within 25% of σtrue . But I am not going to tell you what X
is. You will have to determine X experimentally in lab.
V.
Introduction to Moon Craters
Craters are found on the surface of all the planets and moons in our solar system (except
for the gas giants, which have no solid surface). Until relatively recently, it was thought
that these were the remains of giant volcanoes. That idea began to change around 1960,
but it was not until the manned exploration of the Moon that it was finally confirmed that
the majority of craters were formed by the violent impacts of projectiles from space. It
is now recognized that impact cratering from projectiles as large as ten miles in diameter
had a fundamental influence on the geology of the planets and their moons. In fact, it is
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considered quite likely that the Moon was formed as a result of the Earth’s collision with
a giant asteriod, and that the dinosaurs were killed off as a result of a somewhat smaller
asteriod impact in the vicinity of the Yucatan peninsula.
The purpose of this lab is to study the relation between the size of a crater and the kinetic
energy of the projectile that forms it. Our apparatus will be considerably smaller than
Mother Nature’s: some steel balls and a bowl of sand. You will drop a ball of known mass
m from a height h above the sand and measure the diameter of the resulting crater D. By
varying h and m, it is possible to vary the impact energy E = mgh over almost two orders
of magnitude. Depending on the nature of the mathematical relationship between D and
E—providing you find there is one—you may be able to use your results to estimate what
kind of impact energies produced the kilometer-sized craters observed on the surfaces of
the moons and planets.
Figure 3
VI.
Pre-Lab Exercises
You must have answers to these questions written in your lab notebook at the start of lab.
Your TA will check, and if you don’t, you will be dismissed from lab. It would probably
be a good idea to answer them together with your lab partner (or in even larger groups).
1. Suppose you make 1000 measurements of the same quantity (e.g. the length of a day,
or the lifetime of a radioactive atom) and average them to obtain a best estimate.
Explain how random and systematic errors will each affect the result. How are their
effects similar and how are they different?
2. Suppose you go to lab and measure two quantities D and E. You vary the experimental
conditions so that you have several different values for D with corresponding values
for E. Suppose there is an expected relation
D = bE n
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where b and n are constants. Show how to manipulate your data so that it can be
plotted as a straight line and the exponent n easily determined.
VII.
Procedure
The apparatus for this lab is very simple: some steel balls, a bowl of sand, some measuring
implements, and a few tools to make life easier.
This lab has one unusual aspect: data taking is cooperative. As a class, you will be
dropping four different ball sizes from four different heights. But each group will only be
assigned two different ball sizes and two different heights (for a total of four measurements).
Your TA will assign ball sizes and heights on an arbitrary basis. This way, each group can
take the time to make careful measurements, but you don’t have to stay in lab all night.
There is a chart included at the back of this write-up where you can enter your data and
the data of all the other groups in lab. You will need the full set of data for the analysis.
1. Weigh each ball and record its mass. Be sure to give an uncertainty for each weight.
(This is an example where one careful measurement and one careful check should be
sufficient. Each lab partner should weigh the ball. The balances are precise enough
that you should both get pretty near the same answer.)
2. Plunge the edge of the screen down into the sand several times to loosen it up, then
shake the bowl side-to-side to make the surface level. Do not compact the sand!
3. Drop one of your balls from one of your assigned heights. Be sure to measure the
height from the center of the ball to the top of the sand. Don’t forget to estimate the
uncertainty in your measurement of the height. Keep the impact site near the center
of the bowl, or the crater will not be circular.
4. Measure the crater diameter D. For the purposes of this lab, the diameter is defined
by the very top of the circular lip. Since you are sharing data across groups, make
sure your choice of D is the same as other groups. You can use the small lamp to cast
shadows and more accurately find the top of the crater rim. Use either the calipers
or a ruler. The calipers have what is known as a vernier scale. Make sure you know
how to read it correctly.
5. Repeat steps 2-4 until your results are “good enough.” Once you have repeated the
measurement three times, start computing the standard deviation of your sample.
Note that you have to re-calculate the average after every new measurement before
you can calculate the new standard deviation. When successive computations of σ
are within about 25% of each other, then your results are “good enough.” (Hint: this
should take more than five trials, but way less than 50!)
6. Now use the other ball to obtain a second set of measurements from the same height.
7. Repeat steps 2-6 at your other assigned height. Please note: at some point you will
probably decide that you know how many times to drop the ball without bothering to
track σ after each drop. That is fine. It is even one of the goals of the lab. When you
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reach that point, just go ahead and drop the ball however many times, and calculate
σ once at the end.
VIII.
Data analysis: Part 1
This part of the analysis must be done before you leave lab. There are Macs with Excel
on them, which you may find useful. It would also be a good idea to look over Part 2 and
make sure you understand what’s involved before dashing off.
1. For each data set, calculate the kinetic energy E = mgh of the ball at impact. Be
sure to include an uncertainty. There is an official way to deal with the fact that
E is calculated from two separate measurements with their associated uncertainties,
but we won’t get to that for a while. Do something simple like consider the percent
uncertainty in h (= uncertainty in h divided by h) and the percent uncertainty in
m. Which is larger? Make the percent uncertainty in E the same size as the largest
percent uncertainty in h and m.
2. For each data set, calculate the mean value of D, the standard deviation σ and the
uncertainty (standard deviation of the mean) σm .
3. Enter your values in the chart at the end of the write-up, and fill the chart with the
values from the other lab groups.
IX.
Data analysis: Part 2
First some theory, distilled from the rather extensive literature on crater formation.
After the impact, we wind up with a big hole in the ground. How was it formed? One
possibility is that the energy of the impact went into pushing and reshaping the impact
site, in much the same way as a flat piece of steel is stamped into the shape of a car
fender. This type of process is called plastic deformation. You might then guess that the
impact energy is proportional to the amount of material that gets pushed around. In other
words, the energy should be proportional to a volume, or E ∝ D3 , where we use D just
to give some measure of the amount of material involved. We are not worrying about the
possiblility of some number like 12 or π in front of the D. Tests with underground nuclear
explosions (!) have confirmed this simple idea. (In a deep underground explosion, the
energy has essentially nowhere else to go but into plastic deformation of the surrounding
earth.)
We can make the proportionality more concrete by supposing that an impact of known
energy E0 creates a crater of diameter D0 . Then, in general, the crater diameter formed
from an impact of energy E will be given by
D
=
D0
E
E0
13
.
(3)
A second possibility for crater formation assumes that the energy of impact goes mostly
into throwing material up in the air. (Or up in the vacuum if the target is the Moon. . .)
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In this case, E ∝ M gH, where M is the mass of material ejected, and H is the (average)
height to which is is thrown. But M is a density times a volume. And the volume will be
somehow geometrically related to D (see figure n). We don’t care exactly how its related,
we only need the result that M ∝ D3 . Similarly, the height to which the stuff is lifted will
be related geometrically to D. The result is that in this process E ∝ gD4 . In analogy to
Eq. 3 we can write
14 14
E
g0
D
=
,
(4)
D0
g
E0
where an impact of energy E0 on a planet with gravitational acceleration g0 creates a
crater of diameter D0 . (If all your data is taken on one planet, then that first factor is
just one and can be ignored. But if this simple argument is right, then data taken on, say
Earth, can be used to infer things about craters on, say the Moon. Then you need that
factor out in front to account for the Moon’s weaker gravity.)
Now, which of these equations more closely describes your results?
1. Eqs. 3 and 4 can both be written in the form
D = bE n
(5)
where n is either .33 or .25, depending on the dominant mechanism for crater formation, and b is a constant. Eq. 5 is known as a power law , since D is proportional to
the nth power of E. You have already figured out how to plot an equation of this form
as a straight line. (We will talk more in the next lab about the importance of straight
lines, but the short answer is that a straight line is about the only kind of curve that
can be unambigously identified.) So. . .plot your data in this fashion. (You can use a
spreadsheet program like Microsoft Excel to manipulate and plot your data. There
are Macs in the lab with Excel on them for this purpose. There is also log-log paper
available, which will do the same thing in a more old-fashioned way.)
2. Do the data fall on a reasonable straight line? If so, then they obey a power law and
you can extract the exponent. Find the slope of the line to get n and estimate the
uncertainty in n by eye. (Try drawing different “reasonable” straight lines through
the data and look at the different slopes you get.)
3. Do either Eq. 3 or Eq. 4 describe the results of the experiment? Give an intelligent
discussion of why or why not.
4. The photograph on the next page shows a picture of the Triesnecker crater (named
after a Dead European White Male astronomer who lived from 1745 to 1817) taken by
the Apollo 10 orbiter. If this idea of scaling really works, then you can use the straight
line fit to your data to figure out the kinetic energy of the asteroid that created this
crater, which is 27 km wide. At the surface of the Moon, g = 1.62 m/s2 .
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5. Assume that the velocity of this asteroid was equal to the Moon’s escape velocity
vesc = 2.38 km/s (this is the speed reached by an object falling onto the Moon from
an infinite height). What was the mass of this asteroid? If the density of this asteroid
was typical (2000 kg/m3 ), what was its diameter? Would you expect vesc to be an
over- or under-estimate of the asteroid’s impact velocity?
Appendix A: Useful information
The following equipment is supplied for each lab station:
bowl of sand, cardboard to contain flying sand
screen for un-compacting sand
2-meter stick
two steel balls, ball hold/release mechanism
small plastic ruler, small calipers
incandecent lamp
Macintosh computer, log-log graph paper
The following references on uncertainty and error analysis are on reserve in Kresge.
Taylor, John R.
An Introduction to Error Analysis
Good for the beginner. Doesn’t cover more advanced topics.
Lyons, Lous
Data Analysis for Physical Science Students
Very introductory, but has concise and complete discussions.
Bevington, Philip R.
Data Reduction and Error Analysis for the Physical Sciences
Old fashioned, but a classic. Covers very advanced least-squares methods.
Young, Hugh D.
Statistical Treatment of Experimental Data
More accessible than Bevington, not as friendly as Taylor.
Credits
This lab is based on a crater formation lab first developed by Prof. J. Amato at Colgate
University.