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).
If you used a standard ruler, the smallest marking would be at intervals of .10 cm. If you are able to estimate your measurements accurately to the nearest marking then each ruler location would be accurate to within +-.05 cm.
For example, a marking at 4.7 cm would indicate that the point being measured lies closer to 4.7 cm than to 4.6 cm or 4.8 cm, meaning that the point being measured lies somewhere in the interval between 4.65 cm and 4.75 cm. If a second point is measured at, say, 12.4 cm then this point lies somewhere in the interval between 12.35 cm and 12.45 cm.
For example, a marking at 4.7 cm would indicate that the point being measured lies closer to 4.7 cm than to 4.6 cm or 4.8 cm, meaning that the point being measured lies somewhere in the interval between 4.65 cm and 4.75 cm.
If a second point is measured at, say, 12.4 cm then this point lies somewhere in the interval between 12.35 cm and 12.45 cm.
The distance between 4.7 cm and 12.4 cm is 12.4 cm - 4.7 cm = 7.7 cm. We consider how accurate this distance is.
The first point could conceivably be located at 4.65 cm, and the second at 12.45 cm. The distance between the points could therefore be as much as 12.45 cm - 4.65 cm = 7.80 cm. Similarly the two points might lie at 4.75 cm and 12.35 cm so that the distance between the points is 12.35 cm - 4.75 cm = 7.60 cm. We therefore see that the apparent 7.7 cm distance actually indicates a distance between 7.60 cm and 7.80 cm.
The first point could conceivably be located at 4.65 cm, and the second at 12.45 cm. The distance between the points could therefore be as much as 12.45 cm - 4.65 cm = 7.80 cm.
Similarly the two points might lie at 4.75 cm and 12.35 cm so that the distance between the points is 12.35 cm - 4.75 cm = 7.60 cm.
We therefore see that the apparent 7.7 cm distance actually indicates a distance between 7.60 cm and 7.80 cm.
Our assertion that the distance between the points is 7.7 cm could therefore be interpreted as indicating a distance of 7.70 cm + - .10 cm.
By analyzing the statistical nature of measurement uncertainties we can actually do a little better than this (we can with a reasonable degree of confidence assert that our measurement is +-.07 cm rather than .10 cm), but in this course we aren't going to delve into the statistics. We will take the 'cautious' approach, being sure not to underestimate errors, and say that our measured distance is 7.70 cm +- .10 cm.
In the case of the rubber bands, this assumes that we are in fact able to accurately determine the positions of its ends, accurate to the nearest .10 cm. For carefully made measurements made by individuals with clear vision (whether natural or with the aid of eyeglasses), this is pretty much the case. If measurements are not made very carefully and with good technique, the uncertainty could easily double.
With markings separated by .10 cm, it is actually possible to estimate positions to within .01 cm. However it is not possible to say with confidence that these estimates are accurate at that level.
A practiced individual with very good eyesight and careful technique might be able to reliably make these estimates accurate to within +-.01 cm. For example, a position estimated to be 4.73 cm might be regarded as being accurate to within +- .01 cm, meaning that the actual measurement would lie in the interval from 4.72 cm to 4.74 cm. An analysis similar to that done previously would conclude that the distance between two points so measured is accurate to within +- .02 cm. To measure the length of an actual object with this accuracy, it would be necessary for the points at the ends of the object to be very clearly delineated, with no blurriness or ambiguity in the marking or location of the points.
A practiced individual with very good eyesight and careful technique might be able to reliably make these estimates accurate to within +-.01 cm. For example, a position estimated to be 4.73 cm might be regarded as being accurate to within +- .01 cm, meaning that the actual measurement would lie in the interval from 4.72 cm to 4.74 cm.
An analysis similar to that done previously would conclude that the distance between two points so measured is accurate to within +- .02 cm.
To measure the length of an actual object with this accuracy, it would be necessary for the points at the ends of the object to be very clearly delineated, with no blurriness or ambiguity in the marking or location of the points.
In the measurement of the lengths of the rubber bands with a standard ruler, any estimated uncertainty of +-.02 cm or better should be greeted with skepticism. +_ .03 cm might be acceptable from an observer with a history of very good accuracy in measurements.
Now consider measurements made with a ruler whose smallest markings are separated by, say, .03 cm.
If the user can accurately determine the positions of the two points to the nearest marking on the ruler, then an uncertainty of +_.06 cm is not unreasonable. If positions are estimated to the nearest tenth of a marking, they simply cannot be accurate at that level. A mark between the 8.3 and 8.4 positions on this ruler, estimated at the 8.36 position, probably could not be regarded as accurate to within better than +-.02 units, and possibly no better than within +-.03 units. Converting these results to 'actual' centimeters, we might accept a claim from a good observer that the results are within +-.02 'actual' cm. We would probably be very skeptical of any claim of lesser uncertainty.
If the user can accurately determine the positions of the two points to the nearest marking on the ruler, then an uncertainty of +_.06 cm is not unreasonable.
If positions are estimated to the nearest tenth of a marking, they simply cannot be accurate at that level. A mark between the 8.3 and 8.4 positions on this ruler, estimated at the 8.36 position, probably could not be regarded as accurate to within better than +-.02 units, and possibly no better than within +-.03 units.
Converting these results to 'actual' centimeters, we might accept a claim from a good observer that the results are within +-.02 'actual' cm. We would probably be very skeptical of any claim of lesser uncertainty.
Explain the basis for your estimate of the uncertainty of the length of the first rubber band.
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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 box 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 box), 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 box.
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 box below, in the same order you entered them in the preceding box. Use one line and use comma-delimited format.
In the second line indicate that these results were from the weight of two dominoes.
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 box 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 boxes, 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.
Students typically follow these instructions correctly and provide a good table in the specified format.
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 box 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 box.
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 box.
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.
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 box 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.
Frequently reported descriptions include, but are not limited, to the following:
Increasing at a constant rate then increasing at a decreasing rate.
Increasing at an increasing rate then increasing at a decreasing rate.
Increasing at a constant rate then increasing at an increasing rate.
Increasing at a decreasing rate then increasing at a constant rate.
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 box below.
This would be done by locating the 9.8 cm position on the horizontal axis (the 'L' axis), moving 'straight up' to the curve then 'straight over' to the vertical axis (the 'F' axis). On the above graph, a reasonable estimate might be F = 1.8 N. The estimate you make based on your own graph will of course differ from an estimate based on the graph shown here.
This would be done by locating the 9.8 cm position on the horizontal axis (the 'L' axis), moving 'straight up' to the curve then 'straight over' to the vertical axis (the 'F' axis). On the above graph, a reasonable estimate might be F = 1.8 N.
The estimate you make based on your own graph will of course differ from an estimate based on the graph shown here.
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.
This would be done by locating the 1.4 N force on the vertical axis (the 'F' axis) and moving horizontally to the curve, then straight downward to the horizontal axis (the 'L' axis). A reasonable estimate made on the basis of the above graph might be 9.1 cm.The estimate you make based on your own graph will of course differ from an estimate based on the graph shown here.
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.
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.
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?
There is good reason to believe that the response of your rubber band to increasing stretches is 'smooth', in the sense that the actual force vs. length curve is smooth and that changes in slope occur gradually. It is very unlikely that your data are accurate enough that such a smooth curve can pass exactly through every data point. The deviations of your data points from your smooth curve (assuming you made a good smooth curve according to instructions) are mostly due to measurement uncertainties. A well-constructed curve 'levels out' the inevitable uncertainties in measurement, and tends to be more accurate than any single measurement.
It is very unlikely that your data are accurate enough that such a smooth curve can pass exactly through every data point. The deviations of your data points from your smooth curve (assuming you made a good smooth curve according to instructions) are mostly due to measurement uncertainties.
It is very unlikely that your data are accurate enough that such a smooth curve can pass exactly through every data point.
The deviations of your data points from your smooth curve (assuming you made a good smooth curve according to instructions) are mostly due to measurement uncertainties.
A well-constructed curve 'levels out' the inevitable uncertainties in measurement, and tends to be more accurate than any single measurement.
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?
If your points all lie within +- .05 cm, then it is a pretty good indication that your measurements of length were accurate to within +-.05 cm. You should check your results here against the uncertainties you estimated for your length measurements.