Measurement & Uncertainty CHAPTER 1 Introduction Measurement & Uncertainty Units of Measurement: Quantity Units Abv time seconds s or sec length meters m mass kilograms kg Example: Desk Width 76.74 cm 1 figure estimated 3 figures very certain S1: Chapter 1 Measurement and Uncertainty S3: The Units of Measurement in Physics are similar to those used in Chemistry, but not exactly the same. S5: Time is measured in seconds and abbreviated with a small s or sec. S6: Length is measured in meters and abbreviated with a small m. Abbreviations are case sensitive. S7: Mass is measured in kilograms and abbreviated with a small kg. This is a difference from chemistry where most mass was measured in grams. S13: Almost everything we measure has units associated with it. More importantly perhaps is the fact that almost all numbers used in physics, as in chemistry, are arrived at by measuring something. The numbers that we report to the left of our units of measurement are ONLY those that we were able to accurately measure or at least approximate to a reasonable degree of certainty. For example, when measuring the width of the desk in front of the classroom with an ordinary meter stick we can be very certain that a 7 belongs in the tens place, a 6 belongs in the ones place and a 7 belongs in the tenths place. However, the smallest unit of measurement on the meter stick is .1cm. We can estimate the next place. The hundredths place, but that is the limit of our ability to measure the desk with the measuring device we have chosen—the meter stick. S15: When we can accurately measure three places and estimate a fourth place, we say we have measurement with four significant figures. They are “significant” simply because we were able to measure them. Hundreds ? Tens 7 Ones 6 Tenths 7 Hundredths 4 Thousandths ? Measurement w/ 4 significant figure accuracy

Misc – Prefixes to Memorize M = mega = 106 1000,000 k = kilo = 103 1000 c = centi = 10-2 .01 m = milli = 10-3 .001  = micro = 10-6 .000 001 n = nano = 10-9 .000 000 001 S5: These prefixes are used BEFORE our basic unit abbreviation. They allow us to change vary the number we have measure without changing the meaning of our measurement IF when changing the number we make a corresponding change to the unit by using one of these prefixes. For instance, we could say a soccer field was 100m long or we could say it was .1 km long. Being able to use prefixes correctly and converting from one to another is an important skill. You should also take the time to memorize these prefixes. “Briefly go through the list pronouncing the abbreviation, word, and use of exponents vs use of zeros” S8: In our study of physics a prefix usually indicates a smaller value than the basic unit. This is because we are usually working with relatively small quantities. However, there are just as many prefixes indicating values larger than the basic unit as there are prefixes indicating values smaller than the basic unit. Just look at page 5 of your textbook. NOTE: The prefix usually indicates a smaller value than the basic unit. Example: 1 cm is smaller than 1 m

Misc – Prefixes to Use 1000m = 1 kilometers =1 km 1000,000m = 1 megameter = 1 Mm 1000m = 1 kilometers =1 km 1m = 100 centimeters = 1 x 102 cm 1m = 1000 millimeters = 1 x 103 mm 1m = 1,000,000 micrometers = 1 x 106 m 1m = 1,000,000,000 nanometers = 1 x 109 nm S1: Prefixes are used to create conversion factors. In fact, prefixes ARE conversion factors. Study this list and memorize it for future use. Most of these you should have already learned in chemistry. The new prefixes that you probably were not exposed to chemistry are M (mega), n (nano), and perhaps μ (micro)

Conversion To Proper Units W = 76.74 cm 1 m 100 cm W = .7674 m NOTE: Still 4 S/F Conversion To Other Units (to kilometers) W = .7674 m 1 km 1000 m W = .0007674 km Still only 4 S/F Leading zeros are not “significant” Significant Figures are figures that were measured. S1: We use prefixes as conversion factors all the time in physics. Getting back to the measurement of the width of the desk, it was measured in centimeters and we wish to express that same measurement in meters. Our conversion factor is 1 meter per 100 centimeters. S3: Notice that changing the units results in moving the decimal, but the number of significant figures remains unchanged. Converting from one metric unit to another metric unit should never change our precision (that is, the number of significant figures). S5: Now suppose we wish to convert meters to kilometers. S7: Notice that the same digits still appear in our result; a 7 followed by a 6, followed by another 7 and lastly a 4. However, we now have three zeros between the decimal point and the first 7. We call these “leading zeros” and they are not significant. There are not significant because they do not result from a measurement. They exist only because we changed our units to kilometers. S13: It is obvious in the first example that we have 3 significant figures. The second example is not so obvious. Trailing zeros are significant. If we have a very precise stopwatch that actually displays the tenths, hundredths, thousandths, and ten thousandths place and if this last place displays a zero, then we need to record it in our measurement. The third example show a leading zero to the left of the decimal point. This zero was not measured. Clearly we could write as many zeros over to the left of the decimal point as we wished. However, your textbook often writes a zero to the left of the decimal just so that the decimal appears as a decimal and cannot be confused with a period at the end of a sentence. Other Examples: .435 s 3 S/F .4350 s 4 S/F Record zeros that were measured just like any other measured number 0.4350 s Waste of ink. Not measured, not significant

Conversion To Other Units (to micrometers m) W = 76.74 cm 1 m 1 x 106 m 100 cm 1 m W = 767400 m Place holders are not measured and not significant Still 4 significant figures Question: What if you owned a very expensive and very precise measuring device and there really were zero tens and zero ones. How would you convey the information? S1: If we want to convert from one prefix to another we should do it in two steps and use the conversions factors associated with the two prefixes. S3: In the first step of this process we convert back to the basic unit of measurement. Then in a second step we convert to the new prefix. Notice that our four measured values remain unchanged. However, we now have two trailing zeros in the tens and in the ones place. These are not significant. They were not measured when we took our original measurement. They only appear because we have changed the prefix associated with the basic measurement. We still have only four significant figures. S4: Now the question arises; “But what if I took my original measurements with a very sophisticated and probably very expensive instrument that was capable of measuring increments of 10 micrometers and even capable of allowing me to estimate the number of single micrometers. And further suppose they were both measured and found to be zeros. How would I let the reader of my data to know these values truly were measured (that is, they are “significant“)? S5: A decimal is then used. Answer: 7 6 7 4 0 0 . m Decimal point indicates the “trailing zeros” were measured.

SUMMARY – Test Yourself 340 m 340. m 34000 cm .340 km .34 km .00034 Mm Question: What if the tens are measured and equal zero, but the ones cannot be measured? Answer: 7.6740 x 105 m Through the use of scientific notation we can always convey the proper amount of significant figures. SUMMARY – Test Yourself 340 m 340. m 34000 cm .340 km .34 km .00034 Mm .000340 Mm 2 S/F 3 S/F S1: Another issue could still arise. That is, what if my measuring device could estimate the tens place but was capable of giving me no measurement at all of the ones place. The use of a decimal conveys to the reader that both zeros were measured when one of them was not measured. If no decimal is used, then the reader is left to think that neither zero was measured. How can the correct meaning of my measurement be conveyed to the reader. That is, the zero in the tens place is significant and the zero in the ones place is not significant (not measured) as serves only as a spacer. S3: The use of exponential notation solves this dilemma. We simply show one of the zeros as a trailing zero and we do not show the other zero at all. S5: I hope you correctly answered all of these questions. Let’s review. In problem 1 no decimal point was used and so the trailing zero in the ones place was evidently not measured and is not significant. In problem 2 the decimal point is used and indicates the zero was measured and is significant. Problem three has three trailing zeros that are not significant. Of course the 3 and the 4 are significant. Trailing zeros to the right of the decimal are significant and so problem 4 has three significant figures. Problem 5 has two significant figures (often called “sig. figs.” for short). Leading zeros to the right of a decimal point for a measurement less than and where there are no figures to the left of the decimal point are merely spacers and are not significant. Therefore problems 6 and 7 have two and three sig. figs. Respectively. The trailing zero in problem 7 is the third sig. fig.

Math Operations w/ Sig. Figs. × ÷ Least precise number determines the amount of significant figures in the answer. Example 1: (4.3m/s) (49.317s) = 212. 0631 m per your stupid calculator = 210 m per our S/F rules Reason: We can’t indicate a precision to our measurements that is misleading. S1: When multiplying and dividing two measured values, our answer should only display the number of significant figures contained in the original measurement with the least amount of significant figures. Remember this when doing lab work. If one piece of data is measured imprecisely, then our answer may be imprecise even if all the other data is very precise. S6: In this problem a velocity (measurement with meters per second) was multiplied by time (seconds) to arrive at a displacement (meters). Perhaps the velocity was measured with a radar gun and the time with a very precise stop watch. However, even though time was measured to 5 significant figures velocity was only measured with two significant figures. Therefore the answer could only be properly displayed with two significant figures. We will discuss this more in class. S7: Calculators typically display as many figures as fit in the display. This is often way too many figures and these must be rounded to indicate the proper precision based on our sig. fig. rules. Calculators typically do not show trailing zeros to the right of a decimal point. Occasionally you will have to record a zero that did not show on your calculator. For instance, 6.0/3.0 will result in a display of 2. The correct answer according to our sig. fig. rules is 2.0 (two sig. figs.). Example 2: (34.3m/s) (21s) (476kg) = 1.513235294 m/kg (stupid calculator) = 1.5 m/kg (smart physics student)

Math Operations w/ Sig. Figs. + - Must round the more precise measurement to the same precision as the least Units must be the same Example: l1 = 34.31 cm l2 = 36.9? cm NOTE:  always means 2nd condition minus 1st condition l = 36.9 cm - 34.31 cm S2: Adding and subtracting measure values presents us with another set of issues to deal with. Look at the following example. S5: In this example the initial measurement,l1, the hundredths of centimeters was measured. In the second measurement,l2, the tenths were measured, but not the hundredths. We simply don’t know what this number should be. We cannot simply assume it is a zero any more than we can assume this value is a 6 or a 3 or a 8. We solve this dilemma by rounding the more precise measurement to the same precision as the less precise number. In this case, 34.31 becomes 34.3 and we then subtract as usual. S8: In math class you would get one answer and in physics class you get a different answer. Math is different in that most numbers used in math class are “pure” or “dimensionless” numbers. They have no units and were not the result of any measurements. Math Teacher Answer 2.59 cm Physics Teacher Answer 2.6 cm The math teacher and your calculator assume zeros where none are shown.