Question

A rule of thumb in designing experiments is to avoid using a result that is the small difference between two large measured quantities. In terms of uncertainties in measurement, why is this good advice?

Short answer

In terms of uncertainties in measurement, it is good advice to avoid using a result that is the small difference between two large measured quantities as it can lead to larger uncertainties in the result. When subtracting two measurements, their absolute uncertainties combine directly, which can result in a larger uncertainty as a percentage of the result. This may cause the final result to be less accurate and reliable. To obtain more accurate and reliable outcomes, experimenters should explore alternative methods or measurements that minimize uncertainties.

Step-by-step solution

Understanding Uncertainties

In any experiment or measurement, there is always some degree of uncertainty associated with the results. This uncertainty can arise from various sources, such as limited precision of measuring instruments, human error, and random variations. When calculating a result using multiple measurements, the uncertainties in each measurement can combine and potentially lead to an inaccurate result.

Addition and Subtraction of Uncertainties

Uncertainties propagate through calculations, and the propagated uncertainty often depends on whether the calculations involve addition or subtraction. If two measurements are added, their absolute uncertainties combine in quadrature, which means the absolute combined uncertainty is typically smaller as a percentage of the total than either individual uncertainty. However, when subtracting two measurements, the absolute uncertainties combine directly, which can lead to larger uncertainties as a percentage of the result.

Uncertainty in Subtracting Large Quantities

When using a result that is the small difference between two large measured quantities, the uncertainty of the result becomes more significant. For instance, consider two large measured quantities A and B, with associated uncertainties ΔA and ΔB. The small difference of the two quantities is given by: Result = A - B The absolute uncertainty in the result can be calculated as: ΔResult = ΔA + ΔB If the quantities A and B are large and the difference between them is small, then ΔResult might become a significant percentage of the result, leading to a low accuracy.

Avoiding Results with Larger Uncertainties

To avoid large uncertainties in experiment results, it's important to use a result that isn't a small difference between two large measured quantities. Alternative methods or measurements should be explored to minimize the uncertainties in the result, thus leading to a more accurate and reliable outcome. This rule of thumb helps guide experimenters when designing experiments, as it emphasizes ensuring that the uncertainties in the measurements are minimized while obtaining accurate and reliable results.