The research question for today's experiment is this: What is the frequency of simple harmonic motion for a given mass supported vertically by a spring? See figure 1 below.
Well, one could simply measure the frequency to find out. Ocourse one wants to work with a good model so as to predict that measurement. This lab is about doing both things. Last week we introduced the following over-arching questions about experimental work:
Why do experiments[1]?
In what ways can one use mathematical models to analyze results of measurements?
In this laboratory experiment we engage with the first and second questions. The reference to Franklin's paper suggests several reasons to do experiments, and the ways in which they have been used to make convincing scientific arguments, at least in the subject area of physics. The last section of that paper very briefly discusses the epistemology of experiments, going even further into the ways in which experimental work makes for compelling scientific arguments. As we work through the various tasks described below, begin asking asking yourself what role experiment is playing in supplying a satisfying answer to the research question for this experiment. This experiment is also designed to introduce the use of laboratory notebooks to document our work.
There are no formal lab reports in this class, as we said last time, only lab notebooks [2]. This lab is designed to help us find out about the process of research oriented, professional, experimental documentation. There will be no points given for neatness, or for a particular form of organization. Instead we will focus on (1) learning to take good data, which includes asking and answering the meta-cognitive question, 'how do I know my data is any good?', (2) modeling the data well, which includes it's own meta-cognitive question, and (3) creating good habits for research. These items are interdependent. They suggest deliverables which should go into any reasonable laboratory notebook, which, if well done, will be a hugely important asset for any industrial or academic research lab since they capture ''know-how''. Unless ''know-how'' is well documented, it gets lost as people leave. The lab notebook is kept not just for the experimenter but for the organization that supports the lab. That organization presumably has an interest in the results. As we will see, it isn't just the researcher who needs to read them in the future, but also supervisors and future researchers, maybe even lawyers. The quality of that lab notebook record is judged by what future readers can make of it. The grading rubrics [2] reflect this belief.
As mentioned above, the experimental question for today is this: What is the frequency of simple harmonic motion for a given mass supported vertically by a spring? Experimental work must first be done to determine 'the spring constant', and then simple theory suggests a relationship between that experimental value and the frequency. We will use a models to arrive at a measured and calculated best value for the frequency for a given mass. The model prediction will have some uncertainty arising from the measurement of the spring constant...weird...the theory prediction has uncertainty in it. We will use this to compare with direct measurement, as outlined below. We may be surprised by what we find out!
2.2 Experimental & Modeling Tasks, all jumbled up, so that you meet them as you would if the lab supervisor briefly described the goal, what you had to work with, and then precipitously walked away to a meeting:)
Study a photograph of the apparatus, which is essentially a spring upon which masses may be hung.
Make your own sketch or cartoon of the apparatus in your lab notebook, one that determines everything that what you are measuring (NB1--this is a coding element from the lab notebook rubric, which will appear in various places throughout the description of this experiment, and this experiment only). If there are any important device limitations or quirks, idiosyncrasies, things to be aware of, etc., write these down here. Note that you may not become aware of these until late in the prosecution of the experiment! Leave space. This is a vital part of the record. Also, write down 'the settings' you find necessary for the apparatus (NB2) to serve its purpose, and describe them in your lab notebook [3]. Not some of the settings, all of the settings. Supply all of the crucial information that is initially missing from the sketch above. Supply this in your lab notebook. An odd thing about the term of evaluation, 'crucial'. How do you know what is crucial until you've finished the experiment successfully? So, in the meantime, maybe we adopt the provisional meaning 'all' or 'every bit' to stand in for 'crucial'.
Figure 1. Picture of set up; a mass hanging from a spring. Can you sketch a cartoon of this set up, annotated with nomenclature (variable names) that indicate how stuff is measured? E.g., if there is a position variable, is it clear from the sketch (does the sketch define) what is the reference position?
You will be given a number of small masses, and a spring. Use these as best you know how to get a best value for the spring constant. Then, for a single particular mass, say, for the heaviest mass you try (which could be sum of many small ones), measure the the frequency of oscillation as precisely as you can, and compare this with the frequency you predict by modeling. What is that model?
Write down and explain the mathematical model you have in mind for the way to get the spring constant and the estimation of the frequency of a given mass, how these things could be measured directly, what properties of the spring and the masses hung from it would be important to know, and how they will be found, or measured. Include force laws, invoke functions and constants that make sense to you. It's OK if your understanding of the model is preliminary-it just has to be made explicit. Your aim is to explain to yourself how you are going to solve the problem. You are working in twoes and threes here, so talk it over with your partner. Even if it is only a sentence or two, make this model intelligible to the reader of the notebook (NB1, NB5). Decide on a good heading or headings to go immediately above this entry or entries in your lab notebook: maybe, 'experimental approach', `experimental design', maybe 'modeling approach', maybe, 'how the experiment works'. Put things under headings that draws the readers attention to what you are doing.
Next, prepare a procedure to follow to arrive at your desired result, the answer to the research question for today's experiment. Of course, as you go along, you made need to revise it, but commit yourself to a procedure however provisional, an enumerated list that is your agenda, plan of attack, and so forth. (NB10)
Now begin making measurements. As you do, assess the uncertainty of the things you measure. Justify the size of the 'error bar'. If it's a displacement from some position, define it, explain it, so someone else knows how to do it. If it's a frequency, and all you have is a clock, well, how are you going to do it? What is the uncertainty of your method? Record the rationale in your lab notebook. The size of your uncertainty, rounded up to one significant digit, determines the least significant digit you can include in your measured values [4].
Prepare a table (NB4) with at least 3 columns, $m$, the mass of the hanging mass, $y$, some appropriate measure of position of the hanging mass, and the uncertainty in its position. Note carefully the choice of dependent and independent variables. every column must include descriptors and units, and this description must help the experimenter obtain the desired fitting parameter (or parameters) later on. Quantities and their uncertainties will (shall) have the same units. Some judgment will have to be exercised concerning what they are to be. Suggestion: save room for extra columns (which might occur to you later), one being for comments.
Hang a succession of masses from the spring, recording positions. This is the beginning of a $predict \rightarrow measure \rightarrow compare$ cycle. In succession, hang two masses, letting the total mass accumulate. Before hanging the 3rd mass, do 3 things:
Prepare a plot of position vs. weight or mass in your lab notebook (the quadrille paper makes this easy...) using the first couple of measurements. Label the axes, being clear about units, including them in parentheses on the axes labels. It is easy (common) to forget or ignore them. Make sure your uncertainty is correctly reflected on your graph (NB6) in the form of error bars.
Chose a third mass, and before hanging it
stop and and predict the new position of the hanging masses. This is obviously a modeling step, something often interleaved with experimental steps. Determine what you expect for the new displacement using the model you've indicated. Is your model linear? If you hang twice the mass, is the displacement expected to be twice, or the square root of two, or squared? Which? You have discussed your model above! What kind of model is it? What does it predict? Write down your reasoning. Plot the new predicted position.
Hang the mass and measure the new position. Plot the measurement (remembering error bars). Quantitatively compare the difference between what your prediction was (we are calling this sort of thing a modeling prediction), and the actual measurement showed. That difference is a discrepancy. How does the discrepancy compare with the uncertainty? Write down a sentence or two of how you interpret the comparison. This is a check on progress. It is part of a $predict \rightarrow measure \rightarrow compare$ cycle that you must use to decide whether your data is any good and whether your model is any good.
If your data is 'good', that is, once your data makes sense, then take lots of data, adding masses, filling the table, adding data points on the plot, and so forth. Is a pattern emerging? You will want to add lots of the small masses.
Use Fitteia to prepare a modeling curve to fit the measured displacements. Cut and paste the table directly into the Fitteia data input buffer. Fitteia easily incorporates uncertainties in the dependent variable. Is this choice therefore a reasonable one? Each time there is a modeling task using Fitteia, write down in your notebook an explanation of the computational model you use. On the Fitteia webpage you might write $y = m x +b$, but where can the reader find out, physically, what those variables mean. Wouldn't it be horrible if your 'm' wasn't an $m$ at all, and that your $x$ was an $m$? How confusing is that? Help the reader out!!!
Perhaps your results will look something like that depicted in figure 2. In that case, how is the fitted slope related to the spring constant (the desired parameter)? Explain this briefly. What modeling plots are, and how one make them using Fitteia, are explained elsewhere. But this is another place where your mathematical model is held up next to the data for assessment and comparison (there are $predict \rightarrow measure \rightarrow compare$ cycles big and small in each experiment). Your aim is to make a modeling plot that looks something like the plot below, and please note, in the legend, the expression for the fitting parameter is perfectly hideous and could not be used that way, say, in an abstract, where the correct expressions for significant figures must be made explicit. Uncertainties must, almost always, be rounded up and expressed by a single significant digit. As a result, the best value cannot include more digits than its least significant digit, which is determined the uncertainty. And a parameter that is varied to arrive at the best fit for a model itself has uncertainty, and the same rule applies.
Figure 2. Modeling plot showing some sort of displacement versus some measure of mass. Included in the legend is an expression of one of the fitting parameters, having to do with the slope. Fitteia includes the graph and results in its pdf-report. Note that the values are for purposes of illustration only! The values you measure may differ.
One now has some experimentally derived value for a spring constant. Using the best value for the spring constant, what frequency do you predict for simple harmonic oscillations for one particular mass on the spring. What uncertainty (correctly propagating uncertainties in the quantities that go into the calculation) is there in predicted frequency?
Before doing the last couple of things required to answer the research question, firm up the documentation of the procedure by which you obtain the result, the spring constant \( k \), using Fitteia. Write a 'How-To' that details how to convert the physical model into a computational suitable for modeling (in 'C'), so as to obtain the desired result. Write down the steps you took so that someone else could take the physical model you think applies, the data set (Table 1 above) you acquired, play with a fitting parameter so as to do a 'Goldilocks analysis' of that parameter, and obtain the result (with uncertainty). (NB5, NB8)
Measure the frequency for the given mass described above, detailing carefully how it was done, along with the uncertainty entailed by the method.
Take a breath and reread Fox and Mahanty. Get some coffee. Stroll. Chat with your teammate(s). You have begun the last and most critical of the $predict \rightarrow measure \rightarrow compare$ cycles in this experiment.
One of two things are true now (well, probably there are many things, but right now I can only think two)
You have found good agreement between the model, its frequency prediction, and the experimental results; therefore it is time to write an abstract. See 'notes on lab notebooks' on what to put in an abstract.
You have not found good agreement between the model, its frequency prediction, and the experimental results; something needs to be amended, new ideas need to be explored, and more work is required. It is a matter of some urgency now to read and absorb the import of the abstract, first paragraph of, and figure 1 (and caption) of the paper of Fox and Mahanty [5]. Let's talk it over. Consider the time you have to spend. You will want to get to this point in the work as quickly as possible in case something unexpected is encountered. ( NB: you will want to get to this point in the experiment with enough time remaining so that you can respond to the situation.)
In any case you will finally write an abstract (NB9) quoting your results, comparing uncertainties and discrepancies (expressed with units), using significant figures appropriately, capturing the essence of the methods, and interpreting what conclusions are supported by your work. The abstract will be the last thing you do.
Sorry, the abstract is the penultimate thing you do. The last thing you do is to review your work. Consult the scoring rubric and see if anything is missing, either in the compulsory list of things or in the best practice items of the more holistic evaluative categories.
This lab was written in the pre-pandemic era, and assumed you would be turning in the lab notebook itself for your score for the lab. Rather than surrendering your lab notebook (this has always been impractical) simply create a word document and insert page images of your lab record!!!! This will include of course the Fitteia output! In the old days, one would use scissors and scotch tape...now just save the file and make a screenshot!
References:
A. Franklin, The Roles of Experiment, Physics in Perspective, 1 35 (1999). This paper may be found in Module 2 (SHO w. massive spring) on BlackBoard. Please read, at least, the sections titled Introduction, Epistemology of Experiment, and Conclusions (as an exercise in how to read a paper without having to read the whole paper). Still, I hope you will be beguiled into reading the whole thing. A more philosophically oriented discussion on the same topic can be found (by the same author) in Stanford Encyclopedia of Philosophy (search, Experiment in Physics....read section 2 (Roles of experiment).
NB2 is about relevant settings on equipment and are crucial if one is to understand and later reproduce phenomena and especially, to reproduce the measurements . As these are hard to anticipate ahead of time, one needs to be very keen to write them down as they are used, as the data are being collected, not later during the week when memory fades. Although this set up doesn't seem to have any 'instruments' much less 'settings'. Let me assure you that there ARE relevant 'settings' to write down of a more mechanical nature that are in fact crucial to the measurements! Write them down, note them on any relevant figures.
If in the course of one's experimental work, the uncertainties are themselves well known to correspond to a normal distribution around a mean, and this is saying something about how many times (I don't know, 100) an individual measurement is made, well, then 2 significant figures may be kept in the expression for calculated uncertainties. There are maybe other approaches that justify the keeping of 2 significant figures in an uncertainty, but when one estimates it on the basis of a single measurement, we'll use the approach described, just one rounded up significant digit, even and especially in uncertainties calculated in the process of estimating the value of $\chi^2$. Please refer back to the notes on Goldilocks plots for the estimation of the uncertainty of the best fit value. This is found here.
