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Chapter 1: Problem 103

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 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

01

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.
02

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.
03

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, 螖(A - B) represents the uncertainty in the difference, while 螖A and 螖B represent the uncertainties in the individual measurements A and B, respectively.
04

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.

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Most popular questions from this chapter

You go to a convenience store to buy candy and find the owner to be rather odd. He allows you to buy pieces in multiples of four, and to buy four, you need \(\$ 0.23 .\) He only allows you to do this by using 3 pennies and 2 dimes. You have a bunch of pennies and dimes, and instead of counting them, you decide to weigh them. You have \(636.3 \mathrm{~g}\) of pennies, and each penny weighs \(3.03 \mathrm{~g}\). Each dime weighs \(2.29 \mathrm{~g}\). Each piece of candy weighs \(10.23 \mathrm{~g}\). a. How many pennies do you have? b. How many dimes do you need to buy as much candy as possible? c. How much should all these dimes weigh? d. How many pieces of candy could you buy? (number of dimes from part b) e. How much would this candy weigh? f. How many pieces of candy could you buy with twice as many dimes?

Classify the following as physical or chemical changes. a. Moth balls gradually vaporize in a closet. b. Hydrofluoric acid attacks glass, and is used to etch calibration marks on glass laboratory utensils. c. A French chef making a sauce with brandy is able to burn off the alcohol from the brandy, leaving just the brandy flavoring. d. Chemistry majors sometimes get holes in the cotton jeans they wear to lab because of acid spills.

An experiment was performed in which an empty \(100-\mathrm{mL}\) graduated cylinder was weighed. It was weighed once again after it had been filled to the \(10.0-\mathrm{mL}\) mark with dry sand. A \(10-\mathrm{mL}\) pipet was used to transfer \(10.00 \mathrm{~mL}\) of methanol to the cylinder. The sand- methanol mixture was stirred until bubbles no longer emerged from the mixture and the sand looked uniformly wet. The cylinder was then weighed again. Use the data obtained from this experiment (and displayed at the end of this problem) to find the density of the dry sand, the density of methanol, and the density of sand particles. Does the bubbling that occurs when the methanol is added to the dry sand indicate that the sand and methanol are reacting? Mass of cylinder plus wet sand \(\quad 45.2613 \mathrm{~g}\) Mass of cylinder plus dry sand \(\quad 37.3488 \mathrm{~g}\) Mass of empty cylinder \(22.8317 \mathrm{~g}\) Volume of dry sand \(10.0 \mathrm{~mL}\) Volume of sand plus methanol \(\quad 17.6 \mathrm{~mL}\) Volume of methanol \(\quad 10.00 \mathrm{~mL}\)

The density of an irregularly shaped object was determined as follows. The mass of the object was found to be \(28.90 \mathrm{~g} \pm 0.03 \mathrm{~g}\). A graduated cylinder was partially filled with water. The reading of the level of the water was \(6.4 \mathrm{~cm}^{3} \pm 0.1 \mathrm{~cm}^{3}\). The object was dropped in the cylinder, and the level of the water rose to \(9.8 \mathrm{~cm}^{3} \pm 0.1 \mathrm{~cm}^{3}\). What is the density of the object with appropriate error limits? (See Appendix 1.5.)

Which of the following are exact numbers? a. There are \(100 \mathrm{~cm}\) in \(1 \mathrm{~m}\). b. One meter equals \(1.094\) yards. c. We can use the equation \({ }^{\circ} \mathrm{F}=\frac{9 \circ}{2} \mathrm{C}+32\) to convert from Celsius to Fahrenheit temperature. Are the numbers \(\frac{9}{3}\) and 32 exact or inexact? d. \(\pi=3.1415927\).
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