2.2 Measurement Uncertainties Comparing Results:

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Jacob Carpenter
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1 2.2 Measurement Uncertainties Comparing Results: disregard this section Accuracy and Precision degree of eactness of a measurement like a tight grouping of arrows shot at a target) getting basically the same results ) depends on instrument used it's divisions) Generally, the precision is one half the smallest increment. If an instrument meterstick) measures to the nearest millimeter mm) you would estimate and list the measurement to within at least.5 mm. An instrument like a micrometer can measure to.1 mm and you can estimate within at least.5 mm, therfore, it is more precise than the meterstick. Accuracy and Precision agreement of a measurement with a standard value hitting the bulls eye) two point calibration: checks the accuracy of an insturment by 1) seeing if the instrument reads "" when it should and 2) do it then also give the correct reading of an accepted standard. Techniques of good measurement avoid paralla control outside sources of error heat, ecess pressure) 1

You decide how many divisions you can estimate between the smallest increment the better the instrument the easier it is to estimate).

2 ? Significant Digits tells about instrument and how you use it) Significant digits include all valid digits. This includes all digits up to the smallest increment of the instrument and then one estimated place further. You decide how many divisions you can estimate between the smallest increment the better the instrument the easier it is to estimate). eample: If you used a meterstick to measure the length of your book it would be about.285 m 28.5 cm). The "" digit in the smallest increment of the meterstick the "mm") and the "5" represents the estimated value between the smallest increment. The estimated digit is valid so this measurement has 4 valid digits and therefore 4 sig. figs. 2

repeat Significant Digits Significant digits include all valid digits. This includes all digits up to the smallest increment of the instrument and then one estimated place further.

3 repeat Significant Digits Significant digits include all valid digits. This includes all digits up to the smallest increment of the instrument and then one estimated place further. You decide how many divisions you can estimate between the smallest increment. How you estimate is called Uncertainty of a Measurement and is indicated by a +/ following the listing of the measurement showing how many divisions you divided the smallest increment into. If you thought you could divide the millimeter increment into 1 estimated divisions you would list your tetbook length measurement as.285 ±.1 m. The ±.1 m states you divided the mm increment into 1 divisions are are confident the length is no smaller than.284 m and no larger than.286 m.286 m no larger than.285 ±.1 m.284 m no smaller than Significant Digits With digital instruments like this scale you can assume that all of the digits are significant and it can measure to the nearest pound and estimate to the nearest.1 of a pound.

4 Are all zeros significant? 1) All nonzero digits are significant 2) All final zeros after the decimal point are significant ) Zeros between any two nozero digits are significant 4) Zeros used solely as placeholders are not significant 4

5 Arithmetic with Significant Digits When you use your measurements to calculate additional characteristics/properties by adding, subtracting or multiplying/dividing you have to remember that your results can not be more precise than your measurements. When you add/subtract in a calculation the place of significant digits in the measurements determines the place of significance in the answer. 1.2 mm + 2. mm.5 mm =.5 mm 1.2 mm + 2. mm.5 mm =.5 mm the leftmost place of significance in the addends determine the rightmost place of significance in the answer When you multiply/divide in a calculation the number of significant digits in the measurements determines the number of significant digits in the answer. 2 2 sig. figs. sig. figs 2.2 m 1.24 m m 2 = 2.7m 2 2 answer has to have the least 2) 5

6 Jon has 5 dimes. How many pennies is that? converting 1) 5 dimes ) = pennies units you have units you want 1) write out the problem with the units you have on the left and the units you want on the right...leave room for the conversion factor) 2) determine the conversion factor that is the identify of the two units pennies/dimes) if you put a "1" in front of the largest unit dimes), then the other unit pennies) will always be larger than "1" 1 pennies/1dime) 2) 1 pennies 5 dimes 1 dime ) = pennies Jon has 5 dimes. How many pennies is that? continued units you want 1 pennies 5 dimes = pennies 1 dime units you have ) arrange the units in the conversion factor so the units you have are on the bottom and the units you want are on the top 4) the units you have now cancel dimes on top cancel with dimes on the bottom) and pennies remain 1 pennies 5 dimes = pennies 1 dime 5) multiply the units you have by the conversion factor 5 1= = 5) 5 dimes = 5 pennies 6

7 2 cm = m units you want 2 cm 1 m =.2 m 1 cm units you have 2 cm 1 m =.2 m 1 cm 2 1 = =.2...or 2 1 =.2 _. 1 cm m = cmm)... cmm) cm = m review 1) have/want 2) conversion factor identity ) range, "1" by biggest unit 7

8 = 75 pm Mm pm = Mm prefi pm _1 Mm 1? pm 12 = Mm pm _1 Mm = 18 _ Mm 1 18 pm 12 6 put a "1" 1 ) by the largest eponential separation Mm = 1 6 m pm = 1 12 m eponents: 2 + = ) = 16 Another way to do the same conversion is to identify the eponent values of the measurement and the metric prefies..12 km = m 1) epress the measured value in scientific notation.12 km = km 2) insert your conversion factor with the proper metric units what you have and what you need) km m = km m ) put a "1" or 1) by the largest prefi in the conversion factor km m km 1 = m 4) outside the conversion bracket list the eponent value of the metic prefis) inside the bracket. In this case "m" is a base unit with a value of 1, or 1, hence the, and kilo means 1, or 1, hence, the "" km m = km 1 m 5) list the eponential separation of the metric units in the conversion factor in this case there an eponential separation of meaning 1 or 1) km m = km 1 m 6) This is now the value to insert in the conversion factor m km km 1 = m 7) Because you listed your measurement in scientific notation you can list your primary units in your answer and then and then determine the eponent m km = km ? m 8) determine you eponent by add because you're multiplying) and subtracting because you're dividing) m km = km m = ? m is 1 + which is 2 1 2) 12 1 is 2, which is 2 2 stands for 1 2, that's the eponent in you answer recap: 1 + = 2 8

9 conversion steps 75 pm = mm pm = mm #1: rewrite in standard Scientific Notation #2 conversion factor with what you have on bottom and what you want on top # put a "1" 1 ) by the largest metric prefis you'll never have negative eponents in the conversion factor!!! #4 find eponential separation of your metric prefies #5 This separation is the eponent of your smaller prefi in the conversion factor #6 add subtract eponents! pm 1 mm = mm 1??? pm pm _1 mm = 9 _ _mm pm 12 eponential separation mm = 1 m pm = 1 12 m 2 + = = 7 add because you're multplying subtract because you're dividing speed of what????? _ 11 ft/s = m/s _ 11 ft/s = 1 m m/s.28 ft = 5 9

10 77 miles/hr = m/s miles/hr = m 1 mile no need for eponents in conversion factor when we don't have metric prefies 1 hr 6 s = 4 m/s 1

11 + 15) = = 12 Are all zeros significant? 1) All nonzero digits are significant 2) All final zeros after the decimal point are significant ) Zeros between any two nozero digits are significant 4) Zeros used solely as placeholders are not significant 11

Scientific Measurement

Scientific Measurement Sprint times are often measured to the nearest hundredth of a second (0.01 s). Chemistry also requires making accurate and often very small measurements. CHEMISTRY & YOU How do you