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 summary, using a result that is a small difference between two large measured quantities in experiments is not advisable due to the propagation of uncertainties. The uncertainties in each of the individual measurements combined can lead to a disproportionately large uncertainty in the difference itself. This makes the result less reliable and more prone to inaccuracies. Therefore, avoiding such scenarios helps minimize the impact of uncertainties in measurement, resulting in more accurate and reliable experimental conclusions.

Step-by-step solution

Understanding uncertainties in measurement

Uncertainties in measurement are inherent in any experimental process and are a result of various factors, such as the accuracy and precision of measuring instruments or human error. When interpreting experimental results, it is important to consider these uncertainties to ensure that the conclusions drawn from the results are valid and reliable.

Considering the small difference between two large values

When a result is given as the small difference between two large measured quantities (e.g., A - B, where A and B are large values), the uncertainties in both A and B come into play. This is because the uncertainty in the difference depends on the uncertainties in both of the individual measurements.

The propagation of uncertainties

The propagation of uncertainties is the process through which the uncertainties of individual measurements affect the uncertainty of the overall result. In the case of the small difference between two large measured quantities, we can analyze the propagation of uncertainties through the formula for the difference: \( \Delta(A - B) = \sqrt{(\Delta A)^2 + (\Delta B)^2} \) Here, Δ(A - B) represents the uncertainty in the difference, while ΔA and ΔB represent the uncertainties in the individual measurements A and B, respectively.

Analyzing the impact of the small difference between large values on uncertainty

From the propagation of uncertainties formula, we can see that if A and B are large values with significant uncertainties, then the combined uncertainty in their difference is even larger. Consequently, the ratio of the uncertainty in the difference to the difference itself becomes large, leading to a less reliable result. Essentially, the small difference between two large measured quantities will have a disproportionately large uncertainty, making it difficult to draw accurate conclusions from the result due to higher levels of uncertainty. In conclusion, avoiding the use of results that are small differences between two large measured quantities in experiments is good advice because it helps minimize the impact of uncertainties in measurement. By following this rule of thumb, experiment designers can ensure that the overall uncertainty in the result is kept low, leading to more reliable and accurate conclusions from the experiment.