#$&*
Phy 231
Your 'rubber band calibration' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
#$&* Your initial comment (if any): **
7/25/11
#$&* first line ruler markings, distance in actual cm between ends, how obtained: **
2 hrs
#$&* The basis for your uncertainty estimate: **
Rubber Band Calibration
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In this experiment you 'calibrate' six rubber bands by measuring
their lengths when stretched by varying forces. You will obtain
for each rubber band a table of force vs. length, and you will
construct force vs. length graphs for four of the six bands.
These rubber bands will be used in subsequent experiments.
Most students report that this experiment takes between 2 and 3
hours; some report times of less than 1 hour, some report times
in excess of 4 hours. This version of the experiment defers
analysis of two of the six bands and should require about 15%
less time than the version on which these reports are based.
Taking Data for Calibration:
Note: You should not stretch any of the marked rubber bands
more than 35% beyonds its maximum unstretched length. If you
stretch a rubber band beyond this length you will permanently
distort it. This means, for example, that if a rubber band is 8
cm long you should not stretch it by more than 2.8 cm, to a
maximum length of 10.8 cm.
Important: Throughout the course you will be using the rubber
bands and the calibration graphs you make here, so be sure you
keep the rubber bands and the graphs in a place where you can
locate them, and be sure the graphs are clearly labeled so you
know which one goes with which rubber band.
For this experiment you will use one of the plastic bags that
came with your lab materials and the dominoes from the packet,
along with a ruler, paper clips and marked rubber bands.
You have a bundle of thin rubber bands and a pack of over 100
thicker rubber bands. You will use rubber bands from the pack.
Pick at random six of these rubber bands from your lab kit. If
any of the selected rubber bands have obvious flaws, discard
then and replace with other randomly selected bands. Preferably
using a permanent marker, put 1, 2, 3, 4, 5 and 6 marks on the
respective rubber bands, so you can easily identify them later.
Using paperclips bent into the shape of hooks, form a 'chain' of
all six of your marked rubber bands (a chain of two rubber bands
is shown below). Be sure you observe which is which, and when
you record data make sure that the individual rubber bands are
clearly identified by the number of marks.
Hang the plastic bag from the chain.
Place one domino in the bag.
Measure as accurately as possible the length of the topmost of
your rubber bands. Be sure you keep track of which is which.
Measure from one end of each rubber band to the other. You will
therefore be recording the positions of both ends of each rubber
band. Be sure you measure the end-to-end distance, from the
point where one end of the rubber band ceases and the air beyond
the end begins, to the similar point at the other end.
You should not attempt to align the end of your measuring device
with either of the positions you are recording. Rather align
one of the markings (e.g., the 10.0 cm marking) on your
measuring device with one end of the rubber band, see what
marking corresponds to the other end, and record both markings.
To get the most precise measurement possible you should use a
reduced copy of a ruler. To make sure the measurement is also
accurate, you should take into account any tendency toward
distortion in the corresponding part of that copy. You can
choose whichever level of reduction you think will give you the
most accurate and precise measurement.
In the box below, indicate in the first line the ruler markings
of both ends of the first rubber band, entering two numbers in
comma-delimited format.
In the second line indicate the distance in actual centimeters
between the ends, to an estimated precision of .01 cm..
In the third line explain how you obtained the numbers in the
second line, and what the meaning of those numbers is. Also
indicate how this rubber band is marked, and the limits within
which you think your measurement is accurate (e.g., +- .03 cm,
indicating that you believe the actual measurement to be between
.03 cm less and .03 cm greater than the reported result).
Your answer (start in the next line):
1 cm ,12.83 cm
11.83 cm
For this measurement I used a single reduced ruler. I obtained
my measurement by subtracting the value of one end of the
rubberband from the value of the other end. The rubberband is
marked with 1 mark. I think this measurement is accurate to
+-0.02 cm.
#$&*
Explain the basis for your estimate of the uncertainty of the
length of the first rubber band.
Your answer (start in the next line):
The ruler markings are very close together and it very hard to
be certain which mark the band is closest to.
#$&*
Measure as accurately as possible the lengths of the remaining
rubber bands. Be sure you keep track of which is which. You
may move your measuring device from one rubber band to the next.
In the space below enter the ruler markings of the ends of the
first rubber band, delimited by commas, in the first line (this
will be the same information you entered in the first line of
the last space ), the ruler markings of the ends of the second
rubber band on the second line, etc., until you have a
comma-delimited line for each rubber band.
Then put the word 'End' in the very next line.
Follow this in the very next line by a comma-delimited line
containing the numerical distances in cm, each estimated to
within .01 cm, of the rubber bands in your chain.
Follow this by a line indicating the markings on the rubber
bands.
Finally indicate the uncertainty in your measurements, which
should probably be the same as the uncertainty as that given in
the preceding space .
Your answer (start in the next line):
1 cm ,12.83 cm
1 cm, 12.62 cm
1 cm, 12.48 cm
1 cm, 12.51 cm
1 cm, 12.72 cm
1 cm, 12.75 cm
end
11.58, 11.51, 11.53, 11.51, 11.72, 11.75 cm
1, 2, 3, 4, 5, 6
+-0.02 cm
#$&*
Add another domino to the bag and repeat your measurements. The
positions of the ends should be recorded in your lab book, and
should be backed up electronically in a way you can easily
interpret at any future date (a comma-delimited text file or a
spreadsheet file would be good; a tab-delimited file would also
work but tabs can be variable and invisible so if you are going
to use a text file, a comma-delimited is probably the better
choice).
You won't enter the endpoint information here, but as cautioned
above be sure you have it so if the information reported here
has any anomalies, you can go back to your raw data and correct
them.
Determine the distances in centimeters between the ends of each
rubber band, and enter them in the space below, in the same
order you entered them in the preceding space . Use one line
and use comma-delimited format.
In the second line indicate that these results were from the
weight of two dominoes.
Your answer (start in the next line):
11.63, 11.60, 11.68, 11.72, 11.85, 11.79
The distance the bands streched are from two dominoes in the
bag.
#$&*
Continue adding dominoes and measuring until one of the rubber
bands exceeds its original length by 30%, or until you run out
of dominoes, then stop. To keep the time demands of this
experiment within reason, you should beginning at this point
adding two dominoes at a time. So you will take measurements
for 4, 6, 8, ... dominoes until the 'weakest' of your rubber
bands is about to stretch by more than 30% of its original
length, or until you run out of dominoes.
If one rubber band reaches its limit while the rest are not all
that close to theirs, remove this rubber band from the
experiment and modify your previous responses to eliminate
reference to the data from this band. However, keep the band
and keep your copy of its behavior to this point.
In the space below, enter on the first line the actual lengths
in cm of your rubber bands when supporting four dominoes, in
comma-delimited format. Enter in the same order you used
previously.
On the second line enter the number 4 to indicate that this
result is for four dominoes.
On the third line enter in comma-delimited format the lengths in
cm when supporting 6 dominoes.
On the fourth line enter the number 6 to indicate the six
dominoes being supported.
Continue in this manner until you have entered all your lengths
and numbers of dominoes.
Then on the next line enter 'End'.
You may then enter any brief identifying information or
commentary you wish. However since the nature of the
information has been defined by previous spaces, this is
optional.
If you have reason to believe the uncertainty in your
measurements has changed, indicate this also. Otherwise it will
be assumed that your previous uncertainty estimates apply.
Your answer (start in the next line):
12.10, 12.08, 12.19, 12.18, 12.31, 12.23
4
12.51, 12.32, 12.36, 12.41, 12.42, 12.42
6
12.67, 12.71, 12.73, 12.68, 12.79, 12.67
8
end
#$&*
Compiling and Graphing your Data
Each domino is pulled downward by the Earth's gravitational
field. Each rubber band resists this force by stretching out,
which creates a tension equal and opposite to the force exerted
by the Earth (each rubber band also supports the rubber bands
below it, but the rubber bands don't weigh much so we neglect
that weight). The force exerted by the Earth on each domino is
about .19 Newtons.
Make a table of the force exerted by each of the first four
rubber bands vs. the length of the rubber band. You do not need
to do this with all six, but you should retain the last two
rubber bands and your data for those two, in case you have need
of them in later experiments.
Make a force vs. length table for each of these four bands. The
length will go in the first column, the force in the second.
Your graph will be of the type shown below, but you probably
won't have quite as many data points; your forces will also
differ from the forces indicated by this graph.
There is a tendency for students at the beginning of a physics
course to connect graphs point-to-point. This is a usually a
very bad idea in physics, since there are experimental
uncertainties in our data and we learn nothing by following
those uncertainties around. The graph below is an example of
this Bad Idea.
Note also the REALLY bad idea, which is to treat the 'origin' as
if it is a data point. In this example, we never measured the
force at the 8 cm length, and there is no justification at all
for using the 'origin' as a data point (actually the point
where the axes come together in this graph is not the origin,
it's the point (8 cm, 0); the origin would be (0 cm, 0) and is
well off the scale of this graph ).
It is a good idea to add a smooth curve to the data. This is
because we expect that force will change smoothly with rubber
band length. However we acknowledge that errors might occur in
our data, so we never attempt to make the smooth curve pass
through the actual data points, though we don't try to avoid
them either.
In the example below the curve wobbles around from point to
point instead of smoothly following the trend of the points.
In the next example the curve doesn't try to 'hit' each data
point, but rather to follow the pattern of the actual force vs.
length. It passes among the data points, remaining as smooth as
possible and coming as close as possible to the data points
without making unsightly 'wobbles' in an attempt to pass through
specific data points.
In the space below give your table in a series of lines.
The first line will contain, in the previous order, the lengths
the rubber bands supporting 1 domino, separated by commas,
followed by the downward force exerted by gravity on 1 domino (
i.e., the number, indicating .19 Newtons). You can copy most of
this information (all except the .19) from a previous space .
The second line will contain, in the previous order, the lengths
the rubber bands supporting 2 dominoes, separated by commas,
followed by the downward force exerted by gravity on 2 dominoes.
Again you can copy most of this from a previous space .
Continue in this manner until you have all the lengths and
downward forces, in the same comma-delimited syntax described
above.
Follow your data with a line containing the word 'End'.
In subsequent lines specify the meaning of each column of your
table, the units and the quantity measured in each.
Your answer (start in the next line):
11.58, 11.51, 11.53, 11.51, 0.19
11.63, 11.60, 11.68, 11.72, 0.38
12.10, 12.08, 12.19, 12.18, 0.76
12.51, 12.32, 12.36, 12.41, 1.14
12.67, 12.71, 12.73, 12.68, 1.52
END
Column 1 is for rubberband 1, Column 2 is for rubberband 2,
column 3 is for rubberband 3, column 4 is for rubberband 4.
These measurements are for each length of the corresponding
rubberband using the force for of that set of dominoes. These
measurements are in cm.
#$&*
If you haven't already done so, construct a graph for each
rubber band and fit a smooth curve that you think best depicts
the actual behavior of that rubber band.
In the space below describe the shape of the curve you drew to
approximate the force vs. length behavior of first rubber band.
The curve in the last figure above could be described as
'increasing at a decreasing rate, then increasing at an
increasing rate'. Other possible descriptions might be
'increasing at an increasing rate throughout', 'increasing at a
decreasing rate throughout', 'increasing at an increasing rate
then increasing at a decreasing rate', etc.).
Then describe the shapes of all six rubber bands. Follow your
last description by a line containing the word 'End'. You may
if you wish add comments starting on the next line.
Your answer (start in the next line):
The graph for the first rubberband has a best fit line that
increases at a constant rate. Its origin is about .02 above the
owest point and it extends between the other points ending at
about .02 below the highest point. The graph for the second
rubberband has a best fit line that begins above the lowest
point and runs between the other points ending just below the
last point. This line is increasing at a constant rate. The
graph for the third rubberband has a best fit line that is
increasing at a constant rate and it begins at the lowest point
and runs between the other points ending at about .01 below the
highest point. The graph for the fourth rubberband has a best
fit line that is increasing at a constant rate and begins .01 to
the right of the lowest point and entends between the other
points ending at about .01 to the right of the highest point.
The graph for the fifth rubberband has a best fit line that
increases at a constant rate and begins just left of the lowest
point and extends between the other points and ends at the
highest point. The graph for the sixth rubberband has a best fit
line that increases at a constant rate and begins just above the
lowest point and extends between the other poits ending about
.01 below the highest point.
END
#$&*
Estimating Forces
We can now use our curve to estimate the force at a given
length, or to estimate the length that will give us a specified
force.
In the figure below we estimate the force for the 9.5 cm length.
From the data point it might appear that the force corresponding
to 9.5 cm is about 1.5 Newtons. However we're going to put our
trust in the curve.
We project a line from the L = 9.5 point on the horizontal axis,
straight up to the curve, then straight over to the F axis.
Reading the point on the y axis as F = 2.6 or maybe F = 2.7 we
see that the curve gives us a force between 2.6 and 2.7 Newtons.
If our curve has been drawn carefully and if it appears to make
good sense then we believe that the curve is more reliable than
our data points, and we will tend to believe this estimate more
than our data point.
Similarly we use the curve to estimate the length that gives us
a force of 2 Newtons.
We project a horizontal line from the F = 2 point on the
vertical axis to the curve, then from this point we project
vertically downward to the horizontal axis.
We read a length of about 10.4 cm. Again we use the curve,
which 'averages out' the characteristics of several data points,
to estimate the required length.
If you haven't already done so, include in your report a table
of your data for force vs. length for each of the four selected
rubber bands.
Now for the first rubber band, sketch your best smooth curve,
the one you believe best shows the real force vs. length
behavior of a rubber band. Describe your curve and describe
your thinking about how to construct the curve.
Use your curve for the first rubber band (the one with 1 mark)
to do the following:
Estimate the force in Newtons corresponding to a length of 9.8
cm and report the number in the first line of the space below.
Your answer (start in the next line):
.19 = .4 cm change so 9.8 - 7.2 = 2.6 cm change
and so 2.6 / .4 = 6.5 and 6.5 * .19 = 1.24 N
#$&*
Estimate the length in cm of a rubber band that gives a force of
1.4 Newtons and report the number in the second line.
Your answer (start in the next line):
1.24 = 9.8 so 1.4 = 11.1 cm
11.1 cm
#$&*
From the curve estimate the force in Newtons corresponding to
each of the lengths you actually observed. For example, if you
observed lengths of 8.7, 8.9, 9.3, 9.8, 10.1 cm with 1, 2, 4, 6
and 8 dominoes, what forces would be predicted by the curve for
each of these lengths? Give your estimates in the first line,
using comma-delimited format. In the second line indicate by
how much the estimate of the curve differs from the actual
weight supported.
Your answer (start in the next line):
.19, .38, .76, 1.14, 1.52 N
+-.03 N
#$&*
From the curve estimate, using or your first graph, report in
comma-delimited format, in the first line, the length
corresponding to each of the forces .19 N, .38 N, .76 N, 1.14 N,
etc.. In the second line indicate in comma-delimited format by
how much each of these lengths differs from the length you
actually observed when the rubber band was resisting this force.
Your answer (start in the next line):
11.6, 11.63, 12.1, 12.47, 12.61 cm
0.02, 0, 0, 0.04, 0.06 cm
#$&*
Which do you have more faith in, the values from the curve you
just created or the values you reported in your table, and why?
If you were to estimate a force for a given length using one of
your graphs, what do you think would be the uncertainty in that
force (e.g., +- .12 N, or +- .03 N, etc.) and what is your
evidence for this estimate?
Your answer (start in the next line):
The values from the table are mor acurate because I was able to
see the measurement better.
The uncertainty for me would be about =-.03 N because using a
best fit line that is the farthest away any point of my graph is
from the line.
#$&*
If you were to estimate a length for a given force using one of
your graphs, what do you think would be the uncertainty in that
length (e.g., +- .05 cm, or +- .13 cm, etc.) and what is your
evidence for this estimate?
Your answer (start in the next line):
I think the uncertatinty would be +-.06 based on my previous
calculations.
#$&*
*#&!
&#Very good data and responses. Let me know if you have questions. &#