# adding point (0,0) to data set and its uncertainty?

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 and measure the vertical displacement. There are errors/uncertainties in both the values of the masses used and in the vertical displacement measured. However, I note that when I do not place a mass on the spring there is no vertical displacement. Can I record a measurement of zero mass resulting in a zero vertical displacement? If 'yes', what is the uncertainty in my measurement? If 'no', why not?

There wouldn't be any uncertainty for the mass but the same errors in position would seem to exist for the position of the starting point as exist for the displaced locations of the spring end-point associated with various weights. If you were plotting this, I would think you might have horizontal and vertical "error" bars at all of the weights except the zero weight where instead you might have only a vertical error bar. Will you be measuring the stating point on each separate trial? (I would think you should.)

You might need to be careful in how you analyze this if you are planning a linear model (Hooke's Law) as would seem reasonable. I would not add a zero point in the data, but would rather construct a model with no intercept term. Testing linearity (or determining the elastic limit of the spring) probably would not involve including that point either in my opinion since the displacement measures (differences) would have uncertainties that would include uncertainties in location at either ends.

• Can you clarify/expound on why there would be uncertainty in vertical displacement? If there is no net force applied, and there is an assumed uncertainty in vertical displacement, wouldn’t that contradict Newton’s 1st and 2nd law? Commented Aug 30, 2021 at 21:39
• Our eyeball/brain system can only resolve to about 1/1000 of an inch, and generally my measuring apparatus is not calibrated that low anyway. (I wasn't claiming any quantum fluctuation, just acknowledging the realities of placement and readings of rulers or other distance calibrating devices.)
– DWin
Commented Aug 30, 2021 at 21:48
• So, how would you calculate the uncertainty of displacement? Or would you not calculate at all, and just take the uncertainty in the position of movable end of the spring (the other end is held fixed) as being the uncertainty in the vertical displacement at zero mass? Commented Aug 30, 2021 at 22:34
• The means of estimating uncertainty of displacement under load I know of is $\Delta x = [(\text{uncertainty in reference position})^2+(\text{uncertainty in location with mass} \neq 0)^2]^{0.5}$. But my question is how to determine uncertainty in displacement under load when the load is zero. Is it simply?: $\Delta x = [(\text{uncertainty in reference position})^2+(\text{uncertainty in reference position})^2]^{0.5}$ Commented Aug 30, 2021 at 23:21