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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.
#$&* Your initial comment (if any): **
#$&* first line ruler markings, distance in actual cm between ends, how obtained: **
2hours 30min
#$&* 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):
1cm, 8.4cm
7.4cm
#s obtained by holding a rule behind the rubber band chain as it hung freely and eyeing the marks closest to the ends of the rubber bands. The #s are the stretched length of the rubber bands as they are holding the bag w/ 1 domino in it. They are marked by black marks I put on them indicating their #’s. Measurements are accurate to +-.5cm
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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):
Not being able to hold the chain against the rule without messing up how much each rubber band is being stretched is the big reason for the uncertainty. The other being that I’m merely eyeballing the marks on the rule to the ends.
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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):
1cm, 8.4cm
1cm, 8.4cm
1cm, 8.3cm
1cm, 8.2cm
1cm, 8.2cm
1cm, 8.1cm
End
7.4cm, 7.4cm, 7.3cm, 7.2cm, 7.2cm, 7.1cm
6, 4, 2, 1, 3, 5
+-.5cm
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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):
7.4cm, 7.5cm, 7.5cm, 7.2cm, 7.1cm, 7.1cm
2 dominoes
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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):
7.7cm, 7.7cm, 7.7cm, 7.4cm, 7.4cm, 7.3cm
4
8.1cm, 8.0cm, 7.9cm, 7.7cm, 7.6cm, 7.6cm
6
8.2cm, 8.2cm, 8.1cm, 7.8cm, 7.8cm, 7.8cm
8
8.4cm, 8.4cm, 8.0cm, 8.0cm, 7.9cm, 7.9cm
10
end
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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):
7.4cm, 7.4cm, 7.3cm, 7.2cm, .19newtons
7.5cm, 7.6cm, 7.6cm, 7.3cm, .38newtons
7.8cm, 7.8cm, 7.8cm, 7.5cm, .76newtons
8.1cm, 8.0cm, 7.9cm, 7.7cm, 1.14newtons
8.2cm, 8.2cm, 8.1cm, 7.8cm, 1.52newtons
8.4cm, 8.4cm, 8.0cm, 8.0cm, 1.9newtons
End
Column 1 is the lengths (in cm) of the rubber bands as they are being stretched by the dominoes. Column 2 is the # of dominoes represented by the force applied in newtons (.19newtons = 1dominoe).
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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):
Increasing at an increasing rate, there is little curve to the graph of the line. The higher the force the longer the length of the rubber band.
Increasing at an increasing rate, also little curve to the graph of the line. Slightly steeper than the 1st
Increasing at an increasing rate, climbing faster, getting steeper.
Increasing at an increasing rate, about the same slope as the previous.
End
The graphs of the rubber bands were very similar to each other. As the rubber bands went from the end of the chain that was hanging to the end with the bag of dominoes on it, the tension slacked off. The length of the rubber bands as they went down the chain decreased.
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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):
3.6newtons
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¥ 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):
8cm
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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):
.2, .5, .7, 1
.01, .12, .06, .14
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¥ 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):
7.3cm, 7.5cm, 7.4cm, 7.7cm
.1cm, .2cm, .1cm, 0cm
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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):
Both are close, but the closer one would be the actual data points observed. Because that is what was observed when I did the experiment. The estimated curve of the graph is just an estimate, close enough but not as accurate as the actual data points. +-.2N this is based on the difference of the estimated curve data and the actual points.
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@& If your uncertainties were .5 cm, then your point-to-point graphs would be very jagged as they followed the reading errors from one point to another. The trendline would be much more reliable.
I doubt the uncertainties were that great, the the principle still remains that your data points are more influenced the unavoidable measuring errors than is the overall trendline.*@
¥ 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):
+-.3cm this is also from the difference noted in a previous answer based on actual data points and the estimated curve.
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*#&!
&#Good responses on this lab exercise. See my notes and let me know if you have questions.
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