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Chapter 1: Problem 109

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

Expert verified
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

01

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.
02

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: \(A = 1000mm ± 1mm\) \(B = 995mm ± 1mm\)
03

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: \(Combined Uncertainty = ±(1 + 1) mm = ±2 mm\)
04

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 \(= \frac{2}{5} \times 100\% = 40\%\)
05

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.

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Most popular questions from this chapter

Convert the following Celsius temperatures to Kelvin and to Fahrenheit degrees. a. the temperature of someone with a fever, \(39.2^{\circ} \mathrm{C}\) b. a cold wintery day, \(-25^{\circ} \mathrm{C}\) c. the lowest possible temperature, \(-273^{\circ} \mathrm{C}\) d. the melting-point temperature of sodium chloride, \(801^{\circ} \mathrm{C}\)

Perform the following mathematical operations, and express each result to the correct number of significant figures. a. \(\frac{0.102 \times 0.0821 \times 273}{1.01}\) b. \(0.14 \times 6.022 \times 10^{23}\) c. \(4.0 \times 10^{4} \times 5.021 \times 10^{-3} \times 7.34993 \times 10^{2}\) d. \(\frac{2.00 \times 10^{6}}{3.00 \times 10^{-7}}\)

A column of liquid is found to expand linearly on heating. Assume the column rises \(5.25 \mathrm{~cm}\) for a \(10.0^{\circ} \mathrm{F}\) rise in temperature. If the initial temperature of the liquid is \(98.6^{\circ} \mathrm{F}\), what will the final temperature be in \({ }^{\circ} \mathrm{C}\) if the liquid has expanded by \(18.5 \mathrm{~cm} ?\)

For a pharmacist dispensing pills or capsules, it is often easier to weigh the medication to be dispensed than to count the individual pills. If a single antibiotic capsule weighs \(0.65 \mathrm{~g}\), and a pharmacist weighs out \(15.6 \mathrm{~g}\) of capsules, how many capsules have been dispensed?

For a material to float on the surface of water, the material must have a density less than that of water \((1.0 \mathrm{~g} / \mathrm{mL})\) and must not react with the water or dissolve in it. A spherical ball has a radius of \(0.50 \mathrm{~cm}\) and weighs \(2.0 \mathrm{~g} .\) Will this ball float or sink when placed in water? (Note: Volume of a sphere \(=\frac{4}{3} \pi r^{3} .\) )
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