Instrument resolution and experimental accuracy

Uncertainties in measurements arise from the equipment used and the method employed. As the final paper for this year’s A-level Physics examination is imminent, and tests knowledge of uncertainties, it is worth taking a moment to examine the causes of uncertainty in more detail. We will do this by thinking about two example measurements; one to find the thickness of a sheet of paper and the other to determine the period of an oscillation, such as a swinging pendulum or a bobbing mass on a spring.

The correct instrument to use to measure the thickness of paper is a screw micrometer. Typical laboratory micrometers have a scale marked in divisions of 0.01 mm, giving an uncertainty of +/- 0.005 mm. Knowing how to read a screw micrometer is an expected skill so refresh your memory, if necessary, by reading the post at https://physbang.com/2025/04/09/how-to-read-a-micrometer/ and answering the question sheets I have created on liveworksheets.com (the links are in the article).

Returning to our example, the thickness of a sheet of paper is typically around 0.1 mm but newsprint can be half as thick or even slightly thinner. If we were to use a micrometer to check the thickness of a typical sheet of paper that is nominally 0.1 mm, the uncertainty (+/- 0.005 mm) would amount to +/- 5%. This is not huge but it is significant.

To reduce the uncertainty, we can measure a stack comprising multiple sheets. For example, if we used the same micrometer to measure the combined thickness of 20 sheets, we might get a value of 1.84 mm. Dividing by the number of sheets would give an individual sheet thickness of 0.092 mm. Note that this value is quoted to more decimal places than the micrometer can measure. Not only that but the uncertainty is also divided by the number of sheets, giving +/- 0.00025 mm. So we can now state that a single sheet of this paper has a thickness of;

0.092 mm      +/- 0.00025 mm

Our final percentage uncertainty is therefore +/- 0.3 %, which is much better than can be achieved by measuring one sheet on its own. That said, you might notice the uncertainty figure is greater than would be expected from dividing 5 % by 20: this is because the thickness measurement has decreased slightly, down from 0.1 mm to 0.092 mm.

A similar situation applies when measuring the period of an oscillation except that in this case the uncertainty due to human reactions (when using a stopwatch) is a random error that is likely to be much greater than the limit of the stopwatch’s resolution.

To quote some figures, human reaction times are typically at least 0.15 s whereas the stopwatch can probably display times down to +/- 0.01 s. Note that because stopwatches are (usually) digital, the resolution is equal to the smallest division, not half of the smallest division – as is the case when using analogue devices that are subject to human interpretation.

Once again, by measuring the time taken for 10 complete oscillations, then dividing the total by 10, we can get a value for the period that has just one-tenth the uncertainty that would be incurred if we attempted to time a single oscillation on its own.

Two extra points are worth noting. Firstly, when using light-gates to time the period of an oscillation there are likely to be two signals in each cycle, so to time 10 oscillations we need to get the total time for 20 signals. The reason is that the pendulum (or bobbing mass) will pass through the beam in each direction during one oscillation.

You might think that if the light-gate were put at the very end of the oscillation then that complication would be avoided but in this case any slight loss of amplitude would mean the object falling short of the light-gate and no signal would be generated. To be safe, it is a good idea to put the light-gate at the rest position so there are two distinct signals as the object passes through, one for each direction of travel.

Secondly, things are very different when measuring radioactive decay counts as the events themselves are randomly distributed in time. As a result, the number of counts per minute (cpm) can vary considerably. For typical background radiation, measurements from 20 cpm to 40 cpm are quite common. We could even get some values that fall outside this range – and they would not be anomalies!

Taking measurements over a longer period of time will help to smooth-out the random variations in radioactive decay measurements but it is important to realise that the time period has to be very long, perhaps 20 minutes or even an hour rather than just 60 s.

Finally, as well as checking that you know how to read and interpret a micrometer, you also need the same skills for Vernier callipers. Again, there is a previous post you can read to refresh your knowledge, at https://physbang.com/2025/04/20/how-to-read-vernier-callipers/.

I haven’t written my own test for Vernier callipers but there is a brilliant interactive from Hookean Physics that I highly recommend: you can access it at https://sites.google.com/view/hookean-physics/vernier-scales. After you have entered and checked your answer (be sure to use the required units) just click on Randomise Measurement for the next example.