How do you write an expression of uncertainty of the spring constant by propagation of errors using standard deviation?

Question

A researcher is using Hooke's Law (F= -kx) to determine the spring constant of a spring. The spring is stretched by a force of 2.00 +- .05N and the length the spring changes is measured with a standard ruler to be 25cm.

 

Write an expression of uncertainty of the spring constant by propagation of errors using standard deviation.

 

What is the value of the spring constant, including uncertainty? 

Answer 1

Wow, this is great - someone wanting to do uncertainty instead of calculating error.

High school textbooks all talk about error, %error, accuracy, and precision;

    when in Real Life, it is Uncertainty That Matters.

Even many college textbooks also do that, especially chemistry.

Hurray for you - and your professor.

 

Okay  --

for F= -kx you want the uncertainty of k

where k = - F/x   or ∝ F/x   (the negative doesn't come into play for the uncertainty)

We know the uncertainty of F to be ± 0.05 N

 

(btw when you write a number starting with a decimal point

      Always put a zero before the decimal point like I did above.)

That way if there is a spot on the paper from a copy machine, smudge, etc.

      it will always be clear where the point is.

  If you see a leading zero (ox) on a number you know it is supposed to be o.x

    There is a decimal point even if the copy is bad and you can't see it.

    And if you see a dot but no leading zero then the dot is not a decimal point.

        It is a smudge or spot from a copy machine or something.

 

But we don't know what the uncertainty for X is.

It was measured with a standard ruler - should I assume it had 1 mm markings??

If I do I need to state that.

But still even if it had 1 mm marks, how do I know how careful the person was when they measured it?

Did they measure very carefully to ± 0.5 mm as is possible, or were they sloppy and measured to ± 2 mm ????

Lets assume they measured as well as the instrument (ruler) could, ± 0.5 mm.

    And remember, assume begins with an a, an s, and another s ;)

 

Now:

♣ Order of magnitude is the first approximation.

♦ Significant figures is the second approximation.

♥ Absolute and percent uncertainty is the third approximation.

 

For Additions (and subtractions) we - Add the Absolute uncertainties.

   Note the mnemonic: AAA ,  Additions, Add, Absolute

For multiplication (and division) we - add the percentage uncertainties.

 

Since we have k = F/x we need to find the percent uncertainties and add them.

for F:  we have ± 0.05 N out of 2 N =>> ± 2.5 %

for x:  we have ± 0.5 mm = ± 0.05 cm out of 25 cm =>> ± 0.2 %

and so the maximum uncertainty is ± 2.7 %  ****

 

♠ Data Distribution Curves with deviations are the fourth approximation to uncertainty.

This should only be done when the data analysis and experimental conditions meet formal statistical standards.  Otherwise you will be reporting a sophisticated precision when the conditions do not support it and that sophisticated precision may not actually exist.

If you are designing cars and planes and trains and bridges and stuff;

   people could die if you don't do it right.

 

From the little information given in this example statistical data analysis is Not warranted and should not be done.

But you ask about it; and so as a teaching exercise Only in how to do it, we will try to proceed.

 

1) Repeated Measurements can be made.  (again statistical data should be valid)

    σ_avg = σ / √n

    We have no information on repeated measurements so we cannot do this.

 

2) For addition (and subtraction)

If there is cancellation of errors by some being high and others low (as there will be if the distribution is random and normal) then there is reason to calculate the Root Mean Square (rms) value of the uncertainty.

<< I think the data must be completely independent, sorry can't remember.>>

This can be seen if two distribution curves are plotted and added together, the Gaussian peaks will overlap.

     σ_T = √ ( σ₁² + σ₂² )

This is like absolute uncertainties - you add (but here by rms) the sigmas.

BUT - You are not adding or subtracting.

 

3) For multiplication (and division)

The data must be completely independent.

The formulas for finding the uncertainties are much are a bit more difficult.

It is similar to percent uncertainties where you add percents.

Here you add the percents by an RMS style addition.

     If p = xy or x/y   then σ_p /p = √ [ (σ_x/x)² + (σ_y/y)² ]

 

But, Your data is Not independent!  The Force is related to the stretch of the spring.

- Even if we knew the distributions and standard deviations, which we do not.

This analysis cannot be done.

 

4) There are statistical methods for the standard deviation uncertainties if the data is not independent.

But this is generally much more difficult.  There are papers and textbooks on it.

But my capabilities in this area at this point in my life have decayed, and it is beyond me.

And, I suspect also beyond what you want to do unless you are taking a course in the topic of error propagation and sensitivity.

See:

1) Wolfram Mathworld  - mathworld.wolfram.com/ErrorPropagation.html

2) Sensitivity and Uncertainty Analysis by Dan G. Cacuci

     http://inis.jinr.ru/sl/tot_ra/0/0/3/Cacuci-Sensitiv.pdf

 

 

So the best I can get for the spring constant is:

  k = - F/x = - (2 ± 0.05 N) / (25 ± 0.05 cm)

               =  - (2 ± 0.05 N) / (0.25 ± 0.0005 m) =  - 8.0 N/m ± 2.7 % ****

                      ( **** from the third approximation above)

 

NOW, if when you (or your professor) wrote 2.00 ± 0.05N  what you Really Meant was the measurements had a mean value of 2 N and the standard deviation was ± 0.05 N - then it is another whole ball of wax.

But you didn't specify, so I made no assumptions.

If that was the case you have the σ_F value for the force measurement.  But what about the other?  Still no way to know the σ_x value for the displacement.

 

** I made some edits - they are in Green.