#$&*
PHY 201
Your 'rubber band calibration' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
#$&* Your initial comment (if any): **
#$&* first line ruler markings, distance in actual cm between ends, how obtained: **
5 hours
#$&* The basis for your uncertainty estimate: **
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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):
10.00, 16.67
6.67
The first line contains the ruler markings at each end of the
first rubber band. The second line is the difference of the two
markings. The rubber band is marked “1”. I believe I am
obtaining a level of accuracy to +- .03 cm.
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Explain the basis for your estimate of the uncertainty of the
length of the first rubber band.
Your answer (start in the next line):
I am using a metal tape measure, graduated in millimeters. The
lines are clear enough to estimate approximately one third of the
distance between marks.
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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):
10.0, 16.67
20.65, 27.86
31.86, 39.20
43.21, 50.12
54.20, 61.20
65.18, 71.91
End
6.67, 7.21, 7.34, 6.91, 7.00, 6.73
1, 2, 3, 4, 5, 6
+- .03 cm
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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):
6.74, 7.37, 7.48, 7.02, 7.19, 6.93
The above numbers represent the length of the bands, numbered 1
through 6, in centimeters while supporting the weight of two
dominos.
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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):
6.92, 7.57, 7.72, 7.28, 7.42, 7.07
4
7.02, 7.76, 7.98, 7.45, 7.68, 7.21
6
7.20, 7.95, 8.23, 7.69, 7.69, 7.97, 7.32
8
7.32, 8.13, 8.45, 7.87, 7.87, 8.24, 7.45
10
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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):
6.67, 7.21, 7.34, 6.91, 7.00, 6.73, .19
6.74, 7.37, 7.48, 7.02, 7.19, 6.93, .38
6.92, 7.57, 7.72, 7.28, 7.42, 7.07, .76
7.02, 7.76, 7.98, 7.45, 7.68, 7.21, 1.14
7.20, 7.95, 8.23, 7.69, 7.69, 7.97, 7.32, 1.52
7.32, 8.13, 8.45, 7.87, 7.87, 8.24, 7.45, 1.9
End
Each line above consists of 7 figures in comma delimited form.
The first 6 are the measurements of the numbered rubber bands in
centimeters in consecutive order. The seventh figure is the
force in Newtons.
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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 rubber band number 1 is increasing at an
increasing rate.
The graph for rubber band number 2 is increasing at an increasing
rate throughout.
The graph for rubber band number 3 is increasing at an increasing
rate.
The graph for rubber band number 4 is increasing at an increasing
rate throughout.
The graph for rubber band number 5 is increasing at an increasing
rate then increasing at a decreasing rate.
The graph for rubber band number 6 is increasing at an increasing
rate throughout.
The shapes of the rubber bands were consistent in shape. Bands
numbered 2, 3, and 5 seemed to stretch almost the same, while
bands 1, 4, and 6 did not stretch as much.
End
#$&*
Estimating Forces
We can now use our curve to estimate the force at a given length,
or to estimate the length that will give us a specified force.
In the figure below we estimate the force for the 9.5 cm length.
• From the data point it might appear that the force
corresponding to 9.5 cm is about 1.5 Newtons. However we're
going to put our trust in the curve.
• We project a line from the L = 9.5 point on the
horizontal axis, straight up to the curve, then straight over to
the F axis.
• Reading the point on the y axis as F = 2.6 or maybe F =
2.7 we see that the curve gives us a force between 2.6 and 2.7
Newtons.
• If our curve has been drawn carefully and if it appears
to make good sense then we believe that the curve is more
reliable than our data points, and we will tend to believe this
estimate more than our data point.
Similarly we use the curve to estimate the length that gives us a
force of 2 Newtons.
• We project a horizontal line from the F = 2 point on the
vertical axis to the curve, then from this point we project
vertically downward to the horizontal axis.
• We read a length of about 10.4 cm. Again we use the
curve, which 'averages out' the characteristics of several data
points, to estimate the required length.
If you haven't already done so, include in your report a table of
your data for force vs. length for each of the four selected
rubber bands.
Now for the first rubber band, sketch your best smooth curve, the
one you believe best shows the real force vs. length behavior of
a rubber band. Describe your curve and describe your thinking
about how to construct the curve.
Use your curve for the first rubber band (the one with 1 mark) to
do the following:
• Estimate the force in Newtons corresponding to a length
of 9.8 cm and report the number in the first line of the space
below.
Your answer (start in the next line):
8 N
#$&*
• 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):
7 cm
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• 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):
.18, .39, .75, .95, 1.55, 1.92
#$&*
• From the curve estimate, using or your first graph,
report in comma-delimited format, in the first line, the length
corresponding to each of the forces .19 N, .38 N, .76 N, 1.14 N,
etc.. In the second line indicate in comma-delimited format by
how much each of these lengths differs from the length you
actually observed when the rubber band was resisting this force.
Your answer (start in the next line):
6.63, 6.74, 6.93, 7.05, 7.20, 7.32
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• Which do you have more faith in, the values from the
curve you just created or the values you reported in your table,
and why?
• If you were to estimate a force for a given length using
one of your graphs, what do you think would be the uncertainty in
that force (e.g., +- .12 N, or +- .03 N, etc.) and what is your
evidence for this estimate?
Your answer (start in the next line):
I feel the curve gives a better representation of the data on
the average. Some of my data points seemed to be outside of the
curve I drew. I think I could get within .02 N on most of my
graphs. The curves fall very close to most of the plotted data
points.
#$&*
• 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 would estimate .05 cm. My plotted points seem to vary almost
.05 cm from the curve I used.
#$&*
*#&!
&#Very good data and responses. Let me know if you have questions. &#