[Physics] Uncertainty in measurements with a ruler,

error analysisMeasurements

I am using a 30 cm ruler with a resolution of 0.1cm (1mm). I measure the length of a side of a cube with sharp edges. I keep the ruler against the side of the cube and take two readings from the ruler. Each reading has an uncertainty of ±0.05cm and therefore the measurement will have an uncertainty of ±0.1cm or ±0.10cm? (1st doubt).
The first reading was 0.40 cm and the second reading was 1.25 cm (apparently in the absolute midpoint of 1.20 and 1.30) and the difference is 0.85 cm.
My second doubt is: How can I state the uncertainty?

0.85 ± 0.1 cm (But the estimate and the uncertainty have different sig fig? The estimate can't be stated to hundredths place if we are uncertain about tenths place)

0.85 ± 0.10 cm (But the isn't Uncertainty given to 1 sig fig most of the time, especially in the case of measuring with a ruler of resolution 1 mm)

0.9 ± 0.1 cm (How can we just round it up to 0.9? 1.25 was taken due to the cube ending apparently exactly between 1.20 and 1.30. It does not feel right to me.)

What's gone wrong? Are all of the above wrong? (assume all other factors contributing to error has been eliminated.)

Best Answer

The uncertainty of the measuring instrument is taken to be equal to its least count. This is because when you measure something with the instrument, the mark you read will be the one closest to the actual edge of the object. However, the instrument doesnt allow you to be more precise and hence you may be off by ${\pm}0.1cm$ in case of a standard ruler. Hence your first reading will be $0.4\,{\pm}\,0.1cm$

For your second reading you cannot state it as $0.85cm$ as the least count of your measuring instrument is $0.1cm$. The reading should always be an integral multiple of the least count. You may feel that the mark was right in between $0.8cm$ and $0.9cm$ but you do not know if it is $0.84cm$ or $0.86cm$ or something else.

You can represent the error either as ${\pm}0.1cm$ or as a percentage of the reading ${0.1\over 0.8}{\times}100=12.5$%