Simple Harmonic Oscillation (SHO) with a Massive Spring
In this experiment, you will investigate the simple harmonic motion of a mass suspended from a spring. The primary research question is: What is the frequency of simple harmonic motion for a single, specific, user-chosen mass, supported vertically by a spring?
The lab has two main goals:
Use mathematical models (plural) to predict the frequency of oscillation.
Directly measure the frequency of oscillation and compare it to model predictions (plural).
Conceptual Introduction and Lab Notebook Guidelines
Experimental work allows us to test and refine mathematical models. As you proceed, think about the role of your measurements in either confirming your theoretical model or suggesting a need to adjust it. This lab also introduces you to professional lab notebook practices.
Your lab notebook should serve as a complete and reproducible record of your work. While neatness isn't graded, clarity and thoroughness contrued in a wholistic sense, is. For example, a well-documented notebook should allow another researcher to understand and replicate your experiment. The key deliverables (described in 'Notes on Phys 272 Laboratory Notebooks', which includes the grading rubric) that should be documented include:
NB1: A sketch (block diagram, cartoon, etc.) of your apparatus with all measured variables clearly labeled.
NB2: A detailed list of all equipment settings and any notable quirks or limitations.
NB3: Location of saved data (directory and filename of files created).
NB4: Well-structured data tables with appropriate headings, units, and uncertainties.
NB5: An explanation of your mathematical and computational models. This is called 'How To' in the lab notebook rubric. See Notes on ...Laboratory Notebooks.
NB6: Estimation & Justification of all measurement uncertainties.
NB7: Quantitative comparison of relevant uncertainties and discrepancies.
NB8: Plots with properly labeled axes and error bars.
NB9: A final abstract summarizing your results and conclusions.
NB10: A written procedure or plan of attack.
The experiment involves predicting the frequency of small oscillations of a specific mass bobbing up and down at the end of a vertical spring. Simple physical models leap to mind, 'Hooke's Law' for the spring force on the mass (don't for get the force of gravity), and Newton's 2nd law. From these the frequency can be predicted. Bear in mind the usual simplifications and assumptions entailed when we used these ideas in our previous physics classes (PHYS270, say). How about the following approach: make measurements to determing the spring constant $k$, record the value of the mass put at the bottom of the spring, then, give it a tiny pull and let it go, then use a timer and a counter to measure the period (estimate all the uncertainties, etc. etc.), and arrive at a frequency (with uncertainty, propagated correctly), and compare it with the value that theory (Hooke's law + Newton's 2nd law) assigns to the same thing. Take the difference between the experimental the theoretical values and call that the discrepancy. If the discrepancy is less than the sum of the relavant uncertainties, well, we can say the experimental data is consistent with the model. This sort of experimental approach is an 'agenda', or a 'plan of attack'. What is really is, is a good start. Doing research implies (often) that things stop making sense at some point. Don't panic, but revise and rework the plan. The (whole) record you keep, the 'dairy', is vital to capture true experimental, scientific know-how, which is really what we want to get good at doing. To help with that, read the following.
Experimental Procedure
Study the Apparatus. Examine the spring-mass system shown in Figure 1. In your lab notebook, create a simple sketch of the setup. Label all variables you will measure, such as position and mass. Note any important device limitations, quirks, or settings (NB1, NB2).
Figure 1. The experimental setup: a mass hanging from a spring. Your sketch should define your reference position and other key measurements.
Develop a Mathematical Model. Before you start measuring, write down the mathematical model you expect will describe the relationship between the hanging mass and the spring's extension. Explain how you will use this model to find the spring constant ($k$) and predict the frequency of oscillation. Be explicit about your assumptions. Be sure to distinguish between the computational model and the physical one. Connect and align the two, however, for the reader (this is the heart of the ''How-To'' idea discussed in the rubric for labnotebooks cited above) (NB5).
Prepare a Procedure. Create a step-by-step plan for your experiment. This will serve as your own agenda and can (must) be revised as you go (NB10).
You will need to measure the Spring Constant, probably.
Collect Data. Hang a succession of (very small, smaller than 10g) masses from the spring, starting with two masses. For each mass, measure the equilibrium position and estimate & justify (NB6) its uncertainty. Record your data in a table with columns for mass ($m$), position ($y$), and uncertainty in position ($\Delta y$). Ensure all columns have clear headings and units (NB4).
Iterative Analysis: Predict, Measure, Compare. After your first two measurements, use your data to predict the position for a third, heavier mass. Record this prediction. Then, hang the mass and measure its actual position. Compare your measured value to your prediction. The difference is the **discrepancy**. A discrepancy that is larger than the total uncertainty suggests an issue with your data or model. Document your interpretation of this comparison (NB7).
Complete the Data Set. If your data is "good" (i.e., your measurements and model align), continue adding masses and recording your data until your table and plot are complete.
Analyze Data with Fitteia. Get 'k' as a 'fitting parameter' Use the Fitteia software to create a modeling curve that fits your data. Cut and paste your data table into the Fitteia input buffer. Explain in your notebook what the fitting parameters in the software correspond to physically. For example, if your model is $y = mx+b$, what physical quantities do $m$ and $b$ represent? Your goal is to find the best value for the spring constant ($k$) and its uncertainty.
Figure 2. Example of a modeling plot. Note that the values are for illustration only. You must include your own plot and explain how the fitted slope relates to the spring constant. (NB8).
Measure the Oscillation Frequency. Using a clock of some sort, directly measure the oscillation frequency for a specific, single mass, something small, less than 10g. Call this mass something unique and special (like 'the oscillating mass'). In your notebook, describe your measurement method in detail and justify the uncertainty you assign to it. Write down your results.
Predict the Oscillation Frequency. Using the experimental value for the spring constant ($k$) you found in earlier work, and 'the oscillating mass', calculate the predicted frequency of oscillation for 'the oscillating mass'. Remember to propagate the uncertainty from the spring constant measurement into your frequency calculation. Your *theory estimate* will have an uncertainty too (weird, huh?).
Compare and Conclude. Compare your measured frequency to your predicted frequency. Document your findings (NB7 things) in your lab notebook.
If your measured frequency is in good agreement with your prediction, write a formal **Abstract (NB9)** that summarizes your work, methods, results, and conclusions. OTHERWISE.....
If the values do not agree, (and if you did good work, there will probably be a significant discrepancy), ask yourself: what about the actual spring is NOT modeled by the theory! Consult the provided references to explore why the simple model might be incomplete. By this I mean, read the paper written by Fox and Mahanty. If you have to deal with a significant discrepancy, you are in really, really good company. Most researchers find themselves here! So, then
measure the mass of the spring
determine what fraction of the mass of the spring must be added to the `oscillating mass' in the theory estimate of the frequency. Now the theory includes (what does it include...be clear about this in your report) something it did not before (again, Fox and Mahanty, you can get almost everything you need from its Fig. 1, but you'll have to read and interpret the axes labels...)
recalculate, and return to step 8.
Final Review. The last step is to review your entire lab notebook using the provided rubric. Ensure all required elements are present and well-documented.
Submission. Create a digital document (e.g., a Word file) with page images or screenshots of your lab notebook, including all plots and Fitteia outputs. Save this a *single pdf file*, and submit that on Canvas. The due date will be week's end. These are individual uploads. Obviously there will be signs of sameness in the submissions. The experiment is to be done in groups with lots of conversation surrounding the evolution of the work as it is happening. But each one must create their own diary and do the analysis, etc. for themselves. Groups don't understand things. I know that's an un-nuanced thing to say, but as a mentor and teacher (not to say, grader) I think it....
