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This question assumes only Classical Mechanics.

I am conducting a simple experiment to determine the relationship between the force applied to a spring and the displacement of the spring from its rest length. To do so, I hang various masses from the spring, which produce a force onto the spring, and measure the vertical displacement. There are errors/uncertainties in both the values of the masses (and hence forces) used and in the vertical displacement measured. However, I note that when I do not place a mass on the spring (no force) there is no vertical displacement. I tried this several times and observed the same outcome each time.

Has a precedent been set within the experimental method on recording such an observation? Would a measurement of zero mass (zero force) resulting in a zero vertical displacement be a valid measurement? If 'yes', what is the uncertainty in that measurement? If 'no', why not?

For answers on how to determine the uncertainty (i.e., if 'yes'), we may use the most simple or sophisticated instrument and method for determining uncertainty in displacement of the spring from its rest length. E.g., we may use a marked scale (ruler), estimate position between the scale markings using the method of interpolation, and state the uncertainty based on the ruler's precision. Or we can use high-precision motion sensor (laser) to measure position, make repeated readings and use statistics to evaluate the uncertainty.

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  • $\begingroup$ What do you mean by "Can I [...]?"? That if you do nothing, nothing happens is a useless data point. In a good experiment, you will have so many other data points anyway that the inclusion of this does not matter in the end. But no one can stop you from including it if you really want to. What do you really want to know here? $\endgroup$
    – ACuriousMind
    Commented Sep 4, 2021 at 21:38
  • $\begingroup$ If you want to do honest science, measure the length when you do not add any mass. If it is actually zero, record it. If it is something slightly else, record that. While in theory the spring should be at its rest length, in practice weird things can happen. $\endgroup$ Commented Sep 4, 2021 at 22:02
  • $\begingroup$ You are assuming that the point (0,0) belongs in the data set. Actually put small weights on the spring to verify that this assumption is correct. For some springs, they are "compressed" and show no stretch for very small weights. $\endgroup$ Commented Sep 4, 2021 at 22:09
  • $\begingroup$ @DavidWhite are you suggesting that the data set of (F,x) could be, e.g., (0,0), (1,0), (2,0), etc., so what? $\endgroup$
    – Armadillo
    Commented Sep 4, 2021 at 22:18
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    $\begingroup$ no standards here, you have to measure it a few times and get the error, which, if it never moves, will be the error of your measurement device, as you cannot say that it did not actually move, it might move just less than the error of your measuring device (and of your eye of course) $\endgroup$
    – user65081
    Commented Sep 4, 2021 at 23:14

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Yes, if you actually measured 0 displacement with 0 weight then that is a valid measurement. Make sure that you actually measure it like you measure any of the ones with mass. Follow the exact same procedure with the exact same rigor as your other measurements. Do not skip any step or assume the outcome.

The uncertainty is also treated the same as your other measurements. You may have, in general, two types of uncertainty. One is uncertainty that is evaluated statistically and the other is uncertainty that is evaluated by other means. You should treat the zero measurements the same as your other measurements with respect to both types of uncertainty.

The important thing is consistency with your other measurements. If, for whatever reason, you cannot make the zero measurement using the exact same procedure as your other measurements, then you should not record a zero measurement.

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  • $\begingroup$ For the zero force measurement: I move my hand in same fashion as I would for the non-zero force measurements, the motion of placing a mass on a mass balance to measure its mass. I then make the same motion to attach the zero mass to the spring. Would that count as making a measurement of using zero mass (zero force)? Would that be considered consistent with the non-zero mass/force measurements? Would the uncertainty in this zero mass measurement be the precision of my mass balance? $\endgroup$
    – Armadillo
    Commented Sep 4, 2021 at 23:30
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    $\begingroup$ The motions of your hand is irrelevant. This isn’t a magician’s trick and you are not a mime. If you measure the other masses on a balance then with an empty balance pan write down the measured mass. The point is to not assume that the balance says 0 with an empty pan but to write down whatever it does say. Similarly for length. $\endgroup$
    – Dale
    Commented Sep 4, 2021 at 23:40
  • $\begingroup$ (+1) I use a digital mass balance. I tare (zero) the reading before weighing mass. The empty pan results in a zero mass measurement. $\endgroup$
    – Armadillo
    Commented Sep 4, 2021 at 23:48
  • $\begingroup$ So I tare (zero) my digital balance, find the reading to be zero grams. The precision of the balance is 1 gram (say). So the measurement is 0 $\pm$ 1 grams. But isn’t any negative mass as indicated by the uncertainty range non-physical/non-sensical? $\endgroup$
    – Armadillo
    Commented Sep 5, 2021 at 13:32
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    $\begingroup$ No. If your measurement procedure can produce negative masses (this one can) then that should be recorded. $\endgroup$
    – Dale
    Commented Sep 5, 2021 at 13:58

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