Analisis Experimental

EXPERIMENTAL ANALYSIS
1 UNCERTAINTIES IN MEASUREMENT
Whenever scientists measure a quantity, whether directly or by calculation from more basic measurements, it is necessary to have an idea of how accurate the result is. Some instruments give very accurate readings and the uncertainties in our measurements are negligible (but not zero); other instruments or measuring techniques give substantialuncertainties, and we need to know about this. And in some experiments where we count discrete randomly occurring events (such as decay of radioactive atoms), there is inherent scatter or uncertainty present. Gaining a working knowledge of how to handle uncertainties in measurement (i.e. ‘what to put after the ± sign’) will be an important part of your Experimental Physics course. Knowing theuncertainty in the result of a physical measurement greatly enhances the value of that measurement, for then we know how far it can be trusted. We can make full use of the measurement, without pushing it too far. For example, some estimates give the age of the oldest stars as greater than the time since the Big Bang, which is impossible. Perhaps there are inadequacies in the theory of evolution ofstars and/or the cosmological theory giving the age of the universe. But before reaching that conclusion, it is vital to have good estimates of the uncertainties of the two age values, to see whether the disagreement is really significant. For a more familiar example, suppose you use a ruler to measure the length of a steel rod and find a value of 190.1 mm. You can't be certain of the last figure -maybe the length is really 190.2 mm or even 190.3 (or 190.0 or 189.9). In this case the uncertainty (about 0.1 to 0.2 mm) is clear from the number of figures used to quote the result. We would not give any more because we cannot read them from the ruler. Now you ask someone to calculate what the length will be when the rod has been cut into three equal pieces. The answer comes back as 63.3666667mm. Do you believe that? Quite apart from the limited cutting accuracy and the width of the cuts, clearly those extra figures (...66667) have just been read off a calculator display. Does this matter? Yes, because the extra figures are misleading. They are not valid; they are pure fiction, in fact, and we should report results that are true within their known and stated limitations. Common sense(and a bit of thought) tells you what to expect in the above example with the ruler. The same combination of common sense and consideration of the system you are measuring is the most important ingredient for handling uncertainties in physics. There are also some formulae for calculating and combining uncertainties, which we will deal with in a later section.

Experimental Analysis 8/8/2006

A.11.0 Indicating uncertainty
There are two different ways of indicating the uncertainties in the numerical values of scientific quantities. 1. Explicit - using ± followed by a number. This is often used for results derived directly from experimental quantities. Using the example of the ruler, the steel rod’s length is 190.1 ± 0.1 mm. 2. Implicit - restricting the number of significant figuresso that only the last digit is uncertain. Using the example of the ruler, the steel rod’s length is 190.1 mm. This convention should be used for the final answers in all Physics tests, assignments and examinations. The intermediate steps of any calculation should usually have one or two more significant figures to prevent the accumulation of ‘round-off’ effects. Each of these methods is covered inmore detail below, but first we will clarify the distinction between decimal places and significant figures. 1.0.1 Decimal Places and Significant Figures Decimal places are not the same as significant figures. The following numbers all have 3 decimal places (the number of digits to the right of the decimal point): 1.231 0.100 423.756 0.012 0.003 The following numbers all have 3 significant...