# How do I find the uncertainty?

1. Apr 28, 2014

### boii

I am currently working on my physics lab report, and my question is "What effect does the mass have on the change of acceleration?"

The lab currently consist of two pulleys hanging on each end balancing weighing 100kg, i continuously add 10kg for each turn, i then record the time of the larger mass it takes to reach the table. I am using a digital timer to time each time the weight reaches the table. I am having trouble understanding how to find the uncertainty for my data, as i have barely understanding from it. Here's a picture of my data. And sorry for my lack of grammar as I do try my best. Thanks and much appreciated if anyone could help me!

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Last edited: Apr 28, 2014
2. Apr 29, 2014

### Simon Bridge

Ah - what the engineers call an Atwood machine.
... your data is the measurement of the mass and the measurement of time - presumably also the measurement of the distance fallen?

The uncertainty is the standard deviation of the distribution that would form if you measured the data lots and lots of times - you have to use your understanding of the normal distribution curve to estimate the uncertainty for each measurement you make. You'll have some notes on this - it boils down to a bunch of rules which will seem very wishy-washy at first.

You basically just have to guess how far out your measurements could be about 95% of the time, and halve that number. i.e. when I test my abilities on a stopwatch by timing the swing of a metronome lots of times, I find I can get the values within about 0.12 seconds of each other ... so I figure the uncertainty for me using a stopwatch to be about $\small{\pm}$0.6s.

But when I use a meter ruler to measure lengths, I find I get the same value pretty much all the time. In that case, the uncertainty is always half the smallest division that can be measured. My meter ruler is marked out in mm so my uncertainty on lengths will be $\small{\pm}$0.5mm

Sometimes I am given a measurement - like the mass of a standard weight.
The uncertainty of that value may be stamped on the weight but, if it is not, then just assume that half the smallest sig fig place-value is the uncertainty. i.e. if the mass stamped on the weight is 100g, then the uncertainty is $\small{\pm}$0.5g but if it is 100.0g then the uncertainty is $\small{\pm}$0.05g.

You get the idea?

You should also have some rules about combining uncertainties in your notes.

3. Apr 29, 2014

### boii

Yea I got the notes for that but why would I combine them?

4. Apr 29, 2014

### TheAustrian

What is the equation of the relationship that you are trying to prove?

5. Apr 29, 2014

### boii

f = ma

6. Apr 29, 2014

### TheAustrian

So you need to calculate or estimate your error in your mass (usually around 5% is a good estimate, if your experiment is not of good quality), and estimate your error in the acceleration (for example, your inaccuracy when it comes to using the digital timer, i.e. not being able to stop it right at the precise time)

And then do you know how Gaussian Error propagation works?

7. Apr 29, 2014

### boii

Nope I am afraid I haven't learnt that propagation yet. But how would I find the uncertainty for the timer? Even If I know there is a 5% in error?

8. Apr 29, 2014

### TheAustrian

The uncertainty in the timer is you, the electronics of the timer are assumed to be accurate enough to produce negligible errors.

It takes your brain some fractions of a second to process information. This can vary between 0.1 to 0.3 seconds. It then also takes you some time to press the button on the timer, again this can also vary, depends on how good your reflexes are.

Well, how do you add errors together then?

9. Apr 29, 2014

### boii

Yea I know how to add but how do I know which uncertainties to add and which to leave separate?

10. Apr 29, 2014

### TheAustrian

You need to add all of them. Force is a function of mass and acceleration.
Acceleration is a function of time.

So You must add all of your errors, because Force depends on all of these things.

11. Apr 29, 2014

### boii

So I would multiply the mass and acceleration and add up the uncertainty correct?

12. Apr 29, 2014

### TheAustrian

no, I will explain, wait a few minutes so that I can type this stuff up for you.

13. Apr 29, 2014

### TheAustrian

$F(m, a) = ma$

So your error in mass is $Δm$, and your error in acceleration is $Δa$. Adding these two errors together gives you your error in the Force denoted by: $ΔF$

What you have to do is this:

$ΔF = \sqrt{((\frac{∂F}{∂a})Δa)^{2} + ((\frac{∂F}{∂m})Δm)^{2}}$

where $\frac{∂F}{∂a} = m$

and

where $\frac{∂F}{∂m} = a$

Last edited: Apr 29, 2014
14. Apr 29, 2014

### Simon Bridge

You have to if you want to know the uncertainty on the result of some calculation that includes the measurements.

"Gaussian error propagation" is just a fancy name for combining uncertainties.
You will certainly have this in your notes since you are expected to know how to do it.

I think you have a lot of catching up to do.
You should ask a classmate to take you through that part of the course.

I already gave you examples for estimating errors on your individual measurements.
You will have to relate those examples to your lab-work.

In general, if $x$ and $y$ are independent measurements with uncertainties $\sigma_x$ and $\sigma_y$ respectively, then

if $z=ax$ then $\sigma_z=a\sigma_x$ ... where $a$ is a math constant.
if $z=x+y$ then $\sigma_z=\sqrt{\sigma_x^2+\sigma_y^2}$
if $z=xy$ then $$\sigma_z = xy\sqrt{\frac{1}{x^2}\sigma_x^2+\frac{1}{y^2}\sigma_y^2}$$

You took two measurements for time - do you intend to find the average between the two times or just leave them like that? If you intend to find an average time for each weight, then you will need to find the uncertainty on the average.

You basically need to review your course notes.

Some notes for you:
http://www.met.rdg.ac.uk/~swrhgnrj/combining_errors.pdf
http://www.phys.columbia.edu/~w3081/exp_files/ESDandCombiningErrors.pdf

15. Apr 29, 2014

### boii

Okay got it thanks guys yea my classmates aren't really getting it either we're not learning much from the teacher since its his first year in physic 20 but much appreciated the help

16. Apr 29, 2014

### TheAustrian

What I strongly recommend is that you speak to another lecturer. Ask another lecturer, if he could hold an extra class about error propagation and regression analysis, because it will become more and more important as you progress. I recommend you ask a lecturer that is rather good at mathematics, because error analysis is a very mathematical subject.

17. Apr 29, 2014

### boii

Yea I usually have a tutor in university to help me out but this was a bit of urgent for the lab

18. Apr 29, 2014

### Simon Bridge

Actually fore introductory labs in physics, error analysis is not all that mathematical.
It's usually a matter of learning a bunch of rules with only a very basic overview for the motivation.
Students seldom see the details of the maths behind the rules before postgrad work.

All you need is someone who has done the labs before. A TA should be better for this than a lecturer, but a lecturer is more likely to donate time.

But I agree with The Austrian that this is a serious enough issue to bring to someone's attention.
You want a bunch of you to ask for an extra lesson on error analysis from someone with the authority.
The new guy should know how to do it - but may not realize that there is a problem: never mind it's his first time lecturing - it's not his first time doing. Failing that - go to the year-level dean.
Many colleges have an assistance program where you can get help with homework etc - try the TAs there.

The information you are missing is big enough for 1-2 lessons, so would be out of scope for PF.
Some people here can do distance learning stuff in a pinch - but would normally expect to get paid, so you are better off exploring local resources.

Good luck.