# 9.02 Practical applications

## Example 1

To measure the thickness of a piece of card.
(The true value of the cards thickness is 0.5mm and the measuring instrument has a resolution of 0.1mm)

Problem
The uncertainty due to the resolution of the instrument will be a relatively large fraction of the measurement value (i.e. 0.5mm +/- 0.1mm = 0.5mm +/- 20%).
0.5mm +/- 20% is not very precise!

Techniques

1. Measure the thickness of multiple sheets of card e.g. 20 sheets (which will have a thickness of 10mm). The resolution of the measuring instrument will now be a smaller percentage of the measurement value i.e 10mm +/-0.1mm = 10mm +/- 1% so this is a more precise measurement.

Now to find the to find the thickness of a single sheet we simply divide the measurement by 20. Because the number of sheets (20) is an exact number we divide both the nominal value and the uncertainty by 20 giving 0.5 +/- 0.005 mm which is 0.5 mm +/- 1% (So the greater precision from measuring 20 sheets is maintained) so the thickness for one sheet would be 0.5mm +/- 1%.

2. Use an instrument with better resolution such as a micrometer. This has a resolution of 0.01mm. (Note a micrometer will apply a small amount of pressure to the card when taking the measurement so there is a possibility it may compress it and reduce the thickness. To minimise this, the micrometer has a mechanism that limits the pressure it applies to the object being measured.

Other considerations
It is possible that tiny gaps between the sheets could introduce errors into the measurement. But in practice these will have negligible affects compared to the benefits produced by the improved precision.
It may seem that you have to assume that all the cards have absolutley identical thickness for this appoach to be valid. In fact what you are finding is a nominal card thickness along with an uncertainty interval which will cover any variation between cards