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I am making an uncertainty exercise from a simple Physics experiment based on deriving the equivalent spring constant for a series combination of springs. The mathematical model for the combination of spring is derived. The spring constant of the individual springs, as well as that of the combination is measured by suspending weights.

The uncertainty in the equivalent spring constant from the mathematical model is calculated by transferring the uncertainty in the measurement of the individual springs. This, as I understand, constitutes the uncertainty based on the derived mathematical model. The uncertainty in the equivalent spring constant is also calculated from the experiment.

I am trying to understand how to represent these uncertainties on a graph to justify the validity of the derived model. One way, as I figured, is to take the maximum and minimum uncertainties from the mathematical model and plot two lines (since the model is expected to be linear) for the measurement which can serve as the "bounds" of the mathematical model. This would mean that the mathematical model is valid as long as the experimental measurement lies within this bound. Or that is how I understand it. But I am unable to reconcile the uncertainty in the equivalent spring constant which comes from the actual measurement done by suspending weights.

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  • $\begingroup$ Can you clarify what you've actually measured? Have you put $n=1,2,3,...$ identical springs in series and then measured the total spring constant or have you paired up a variety of 'known' springs and measured their effective spring constants? $\endgroup$ – jacob1729 Sep 1 at 15:37
  • $\begingroup$ Hi. Thanks for your reply. The setup has two springs of different, unknown spring constants. A theoretical model is first derived for the effective spring constant (equivalent spring constant) of the series combination of the two springs. The spring constant of the individual springs is measured together with the uncertainty in the measurements. These uncertainties are then transferred to the equivalent spring constant as per the model. Hope this helps. $\endgroup$ – Shaz Sep 1 at 18:19
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Furnishing a plot with an upper and lower bound line showing the potential maximum errors possible is indeed one way of visualizing this. The more common way however is to replace each individual data point with a vertical line that spans the range from the upper error bound to the lower error bound. The midpoint of the line is the value of the data in the absence of errors. These vertical lines are called error bars.

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  • $\begingroup$ Thanks for your reply. That can work too, but I think that there would error bars with the experimental measurement of the equivalent spring constant. I believe that there would be too many error bars for the student to decipher the information. So, I am wondering if adding too many error bard would make the graph look complicated. $\endgroup$ – Shaz Sep 1 at 18:25
  • $\begingroup$ no, what you do is include ALL error and uncertainty sources into ONE error bar. $\endgroup$ – niels nielsen Sep 1 at 18:26
  • $\begingroup$ But there are two sources. 1) The equivalent spring constant determined from the derived mathematical model has uncertainty because the individual spring constants are uncertain. 2) When I combine the springs and measure their equivalent spring constant, that measurement is unsure because of the inherent uncertainty in measuring mass and length and also the "quality" of the linear fit. These uncertainties in the measurement are something I can combine to put total error bars. But as I understand, the uncertainty from the mathematical model and that coming from the experiment are different. $\endgroup$ – Shaz Sep 2 at 3:27
  • $\begingroup$ you'd specify that the error bar includes uncertainty in the spring constant measurement. this is the usual thing. $\endgroup$ – niels nielsen Sep 2 at 4:04

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