What is the best way to estimate uncertainties in a lab experiment?

[yolo123]

I am asked to indicate the uncertainty on the radius R of rotation of the plumb bob. Please see picture.
Basically, there is a vertical shaft, over which is attached a horizontal shaft and a plumb bob is hung over with a thread on the horizontal shaft. The bob is attached with a spring to the vertical shaft. In lab, we start rotating the vertical shaft until the spring is stretched and the bob's end is just over a metallic indicator (a bar).
The diameter of the indicator was 0.3 cm.
We did this lab and my teacher corrected us. But he accepted different answers.
My friends just put a 0.1 cm uncertainty on the measurement on R. THAT'S IT.
What I did is that I estimated my pinpointing of the center of the vertical shaft by +/-0.1cm. Then, I added 0.3cm (diameter of indicator). TOTAL UNCERTAINTY : 0.4 cm

Which one is RIGHT?

 

[Simon Bridge]

You only ever estimate uncertainties.
The estimate should be the smallest value you think of that you can argue is guaranteed to be bigger than the actual uncertainty.

Consider - you have some reference position you want to call r=0 - the origin.
You put an indicator at position r=R and it has diameter d
The bob is at position r ... in the course of the experiment you change r so that it sits as close to the middle of the indicator as you can get it before losing patience.

Now you measure r.

The uncertainty on r depends on how you did the measurement.

For instance:
1. You got a tape-measure marked out in millimeters and just measure from the origin to the point of the bob.
That would get you $\small\pm$0.5mm if you were very careful with the way you placed the origin end of the tape.

This is usually the default where calibration has been careful. The uncertainty on a single measurement is estimated to be half the smallest interval on the scale.

2. As with 1 except you just lay the tape next to the setup and take a reading of a for the origin and b for the bob.
This gets you r=(b-a)$\small\pm$1mm
... because both a and b are uncertain to 0.5mm

3. You used the tape to measure R, and are happy to use that as r as well. Saves effort after all.
Then you could say that r=R$\small\pm$(d/4). [or d/6 depending on how confident you are]

For you that means an uncertainty of 0.8mm if you are very confident and 0.5mm if you are almost certain that the bob is smack-on the center of the indicator. (the 4 is 2sd either side of the center and the 6 is 3sd either side.)

4. As with 3 except the indicator surface is marked out in millimeters. You record x millimeters from the center of the indicator
So you can say that r=(R+x)$\small\pm$1mm
... because both R and x are uncertain to 0.5mm.

5. You take a large number N : N>10 measurements of r by any of the above methods or some other method, resetting the measuring equipment each time. This gives you a set of r values.

Take the actual r value as the average of the set, then the uncertainty is 1/√N times the standard deviation of the set.

6. As with 5, but there is no variation in the measurement!
Then compute the error according to the method and divide it by √N.

You can see there is something of an art to this.

Uncertainties are almost always quoted to 1sig.fig. so most of the complicated way end up the same as a simple one after you get used to it.

If you work through these for your problem, I suspect that most of the different approaches amount to ##\small\pm#1mm.