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Worksheet 1: Uncertainty analysis

In this first lab we reinforce the concept of uncertainties based on simple measurements of length and volume of objects and their masses with different tools. Everything we measure has uncertainties, and to be able to say whether a measurement can be reproduced or agrees with theoretical predictions, we need to carefully take them into account.

It is important to distinguish between statistical/random and systematic uncertainties. They are fundamentally associated with the terms of precision and accuracy. Statistical uncertainties can be reduced by repeated measurements up to a degree, while systematic uncertainties require improvement of measurement procedure or measurement devices. See this introduction for more details.

The standard deviation of a single measurement describes its average uncertainty. The average of multiple measurements is taken as the best estimate, and its reduced uncertainty is given by "standard deviation of the mean". When we measure individual quantities with uncertainties, and those quantities enter through a mathematical formula in a derived quantity, we use uncertainty propagation to compute the uncertainty of the derived quantity. For example when we measure the volume with a ruler, each dimension measurement will have an uncertainty, which needs to be propagated in the volume V=w\cdot h \cdot l with width, height and length measurements for a rectangular object.

Measurement 1

For measurement 1 you use the rectangular block and the washer with a hole in the center shown in figure 1. The measurements are done using a transparent plastic ruler as in figure 2.

Figure 1: Left: Rectangular block, right: metal washer with hole in center.
Figure 2: Top: metal washer with hole in center. Bottom: transparent plastic ruler with a systematic read-off uncertainty of 0.5\,\text{mm} (half a division).

Measurement 2

Many measurement devices require calibration. For example calibrating a scale means that we test it with weights of known mass, and set its display's response to the mass in expectance with the known masses. Calibration may be required for many reasons. Further, for example, the scale should show a measurement of zero when no weight is measured, even though there is usually a baseplate of a certain mass on top of a spring-operated scale.

The calibration in measurement 2 is done using the wooden ruler in figure 3.

Figure 3: Top: wooden ruler. Bottom: transparent plastic ruler. Both rulers have a systematic read-off uncertainty of half a division.

Measurement 3

For measurement 3 of masses and densities, we use the triple beam balance in figure 4, the digital scale in figure 5, and the calibration weights in figure 6.

Figure 4: Triple beam balance. The systematic read-off uncertainty is 0.05\,\text{g} (half a division).
Figure 5: Digital scale with a systematic uncertainty of 1\,\text{g} (1 digit).
Figure 6: Calibration weights of 500\,\text{g} down to 10\,\text{g}. We assume these to have negligible uncertainty.