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 measurement uncertainties, avoiding small differences between large measured quantities is good advice because the uncertainties from each measurement can combine, resulting in a larger overall uncertainty for the final result. This can greatly affect the overall accuracy and reliability of the result and may lead to misleading conclusions. Instead, it is better to measure quantities directly or design experiments such that the difference between comparable quantities is not too small compared to their uncertainties.

Step-by-step solution

Observe Measurement Uncertainties

When measuring a quantity, there will always be some level of uncertainty associated with the measuring process. This uncertainty can come from various sources such as the limitations of the measuring instrument, human error, or external environmental factors. Uncertainties are typically expressed using a range or an error value, indicating how much the actual value could deviate from the measurement.

Analyze the Difference Between Two Large Measured Quantities

When comparing two large measured quantities with uncertainties, the uncertainties of each measurement combine and may result in a larger overall uncertainty for the final result. Consider the following example: Let's say you have measured two distances, A and B, with uncertainties of ± 1mm. The measurements are: \(A = 1000mm ± 1mm\) \(B = 995mm ± 1mm\)

Calculate the Difference and Combine Uncertainties in the Result

Now we need to find the difference between A and B, which is a small value. We also need to combine the uncertainties related to each measurement: \(Difference = A - B = (1000 - 995) mm = 5 mm\) The combined uncertainty is the sum of individual uncertainties: \(Combined Uncertainty = ±(1 + 1) mm = ±2 mm\)

Assess the Resulting Uncertainty in the Difference

We can now observe how the combined uncertainty affects the difference between these two measurements. The difference we found in the previous step is 5mm, with a combined uncertainty of ±2mm. This means that the actual difference could be anywhere between 3mm and 7mm: \(Difference = 5mm ± 2mm\) In terms of percentage uncertainty, this represents a relatively large value: Percentage Uncertainty \(= \frac{2}{5} \times 100\% = 40\%\)

Understand the Importance of Avoiding Small Differences Between Large Measured Quantities

From the example above, it is evident that when dealing with small differences between large measured quantities, the combined uncertainties from each measurement can greatly affect the overall accuracy and reliability of the result. By avoiding such situations in experimental design, we can minimize the chances of getting misleading or inaccurate results due to compounded uncertainties. In general, it is better to measure quantities directly or to design experiments such that the difference between comparable quantities is not too small compared to the uncertainties in their measurements.

Key concepts

Experimental Design

When setting up an experiment, careful consideration is necessary for what measurements to take and how they will be taken. The goal is to ensure that the results will be both accurate and reliable.

• One common guideline in experimental design is to avoid situations where the result is a small difference between two large values.
• This is because any measurement has a degree of uncertainty, which can significantly skew the result when differences are small.
• Designing experiments with this in mind helps in minimizing errors that could lead to misleading results.

In practice, this means measuring directly whenever possible or structuring the experiment so that the uncertainties do not overshadow meaningful differences. This approach can improve the overall integrity of the experimental results.

Combined Uncertainties

Measurements are never without error. Each measure has its uncertainty, which represents how much the measurement could vary from the actual value.

• When working with multiple measurements, you have to consider the combined uncertainties.
• For example, if measuring two distances, each with their uncertainties, these must be accounted for in the result.

The combined uncertainty is calculated by summing the individual uncertainties, as shown in the earlier exercise:\[Combined \ Uncertainty = \pm (1 + 1) \ mm = \pm 2 \ mm\]Understanding this concept helps ensure that results, especially those involving differences, are as accurate as possible.

Percentage Uncertainty

Understanding uncertainty can often be more intuitive when expressed as a percentage of the measurement.

• Percentage uncertainty offers a way to represent uncertainty relative to the measured value.
• The formula for calculating percentage uncertainty is:\[Percentage \ Uncertainty = \frac{Absolute \ Uncertainty}{Measured \ Value} \times 100\%\]
• Applying it to a result helps illustrate how significant the uncertainty is in context.

In the exercise example, the percentage uncertainty was calculated as:\[40\% = \frac{2 \ mm}{5 \ mm} \times 100\%\]This high percentage indicates that the uncertainty constitutes a significant portion of the measured difference.

Measurement Accuracy

Accuracy in measurement is crucial in experimental settings. It dictates how close a measurement is to the true value.

• Inaccuracy can come from both random errors and systematic errors inherent in the measurement process.
• Knowing the level of uncertainty influences how accurate a measurement is perceived to be.

When measurements produce small differences relative to their uncertainties, like the 5 mm difference with a ±2 mm uncertainty in our example, the accuracy becomes questionable. Therefore, precision in measurements and understanding their uncertainties is key to achieving accurate experimental outcomes. Balancing accuracy with the reality of measurement errors is a critical part of scientific inquiry.