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CHAPTER 2

CHAPTER 2. The Metric System Conversions Measurement Significant Digits Graphing. Measure The Room Lab. Measure the length and width of the room in “shoe” units. Discuss differences. Why doe we use standards? Name US units volume, distance, …etc Name Metric “Base” units.

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CHAPTER 2

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  1. CHAPTER 2 The Metric System Conversions Measurement Significant Digits Graphing

  2. Measure The Room Lab • Measure the length and width of the room in “shoe” units. • Discuss differences. • Why doe we use standards? • Name US units volume, distance, …etc • Name Metric “Base” units. • Name Metric prefixes • Place prefixes in order smallest to largest.

  3. 2.1 The Metric System and SI • Why use the Metric System • Based on powers of 10, convenient to use • The Syste’me Interationale d’Unite’s = • SI = The Metric System • This is the standard system used throughout the world by scientists, engineers, and everyone else everywhere except US.

  4. Why don’t we use the Metric System in the United States? • Good question.

  5. Base Units • Length = meter (m) • Mass = gram (g) → “standard” unit = kg • Volume = liter (l) • Time = second (s) • Temperature = Kelvin (K) • Amount of a substance = mole (mol) • …etc

  6. Definition of a kilogram • The mass of a small platinum-iridium metal cylinder kept at a very controlled temperature and humidity.

  7. Definition of a meter • The distance traveled by light in a vacuum during a time interval of 1/299 792 458 seconds.

  8. Definition of a second • The fraquency of one type of radiation emitted by a cesium-133 atom.

  9. SI Prefixes • Giga – G 109 1 000 000 000 • Mega –M 106 1 000 000 • kilo – k 103 1000 • hecto – h 102 100 • deka – da 10 10 • BASE UNIT meters/liters/grams/…etc • deci – d 10-1 0.1 • centi – c 10-2 0.01 • milli – m 10-3 0.001 • micro – μ 10-6 0.000 001 • nano – n 10-9 0.000 000 001 • pico – p 10-12 0.000 000 000 001

  10. Introduction to meter stick • Meter (m) – dm – cm - mm

  11. METRIC “STEP” SYSTEM & CONVERSIONS • Convert 102m → mm • 102m = 102 000mm • Convert 427 693μm → m • 427 693μm = 0.427 693m • Metric Conversion Hand Out #1

  12. METRIC “Step System” G G G M M M k k k BASE BASE BASE h h h da da da d d d c c c m m m n n n p p p For each step you go up, move the decimal point one place to the left. For each step you go down, move the decimal point one place to the right. μ

  13. Conversions Cont. • Length 1 inch = 2.54 cm (exactly) • Volume 1 liter = 1.0576 qt • Mass 1 kg = 2.21 lbs • Weight 1 lb = 4.45N (Newtons) • More useful conversions on back cover of text book

  14. Dimensional Analysis Method= Factor Label Method • Example #1 • 4km → in • 4km x 1000m x 100cm x 1in = • 1 1km 1m 2.54cm • 157 480in

  15. Example #2 • 26dam → yds • 26dam x 1000cm x 1in x 1ft x 1yd = • 1 1dam 2.54cm 12in 3ft • 284.3yds

  16. More Example Problems • 3) 37hl → gal • 37hl = 978.28gal • 4) 439 672 107mg → tons • 439 672 107mg = 0.4858tons

  17. 5) 467 223 921 732 oz → Gg • 467 223 921 732 oz = 13,269.16Gg • 6) 937 456 737mg → tons • 937 456 737mg = 1.03tons • Hint: • Insert the units first to ensure that the units will cancel out leaving only the unit that you want to end up with.

  18. Handouts #3 & #4 • When using the dimensional analysis method it is very helpful to insert units first, then the proper numbers.

  19. SCIENTIFIC NOTATION(and Calculators) • Convert 276Gl → pl • 276Gl = 276000000000000000000000pl • Convert 146ng → Mg • 146ng = 0.000000000000146Mg

  20. Q: Is it convenient to use these types of numbers? • A: NO!!!!!! • Scientific Notation is used to represent these very large/small numbers.

  21. Rules for Scientific Notation • The numerical part of the quantity is written as a number between 1 and 10 multiplied by a whole-number power of 10. • M = 10n where: 1 ≤ M < 10 n is an integer

  22. If the decimal point must be moved to the right to achieve 1 ≤ M < 10, then n is negative (-). If the decimal point must be moved to the left to achieve 1 ≤ M < 10, then n is positive (+). 100 = 1

  23. Therefore written in proper scientific notation: • 276000000000000000000000 pl = 2.76 x 1023pl • 0.000000000000146ng = 1.46 x 10-13Mg

  24. Calculator Buttons • In class examples of E, EE, and positive/negative exponents.

  25. Addition & Subtraction • If the numbers have the same exponent, n, add or subtract the values of M and keep the same n. • 3.7 x 104 + 6.2 x 104 • = (3.7 + 6.2) x 104 • = 9.9 x 104

  26. Example-2 • 9.3 x 107 - 4.1 x 107 • = (9.3 – 4.1) x 107 • = 5.2 x 107

  27. If the exponents are not the same, move the decimal point to the left or right until the exponents are the same. Then add or subtract M. • Example-1 • 2.1 x 108 + 7.9 x 105 • = 2.1 x 108 + 0.0079 x 108 • = (2.1 + 0.0079) x 108 • = 2.1079 x 108 • or

  28. Example – 2 • 2.1 x 108 + 7.9 x 105 • = 2100 x 105 + 7.9 x 105 • = (2100 + 7.9) x 105 • = 2107.9 x 105 • = 2.1079 x 108 • Exactly the same as previous example

  29. If the magnitude of one number is very small compared to the other number, its effect on the larger number is insignificant. The smaller number can be treated as zero. (9.99 x 103 = 9999) • 7.98 x 1012 - 9.99 x 103 • = 7980000000 x 103 - 9.99 x 103 • = (7980000000 - 9.99) x 103 • = 7979999990.01 x 103 • = 7.98 x 1012

  30. Multiplication • Multiply the values of M and add the exponents, n. Multiply the units. • 4.37 x 107m x 6.17 x 1013s • = (4.37 x 6.17) x 10 (7 + 13) (m x s) • = 26.9629 x 1020ms • = 2.69629 x 1021ms

  31. Division • Divide the values of M and subtract the exponents of the divisor from the exponent of the dividend. Divide the units. • 7.9 x 109 m4 • 3.1 x 106 m3 • 7.9 • = 3.1 x 10 (9 -6) m (4-3) • = 2.548 x 103m

  32. Challenging Addition • 8.9 x 105m + 7.6 103km • = 8.9 x 105m + 7600 x 103m • = 8.9 x 105m + 76 x 105m • = (8.9 + 76) x 105m • = 84.9 x 105m • = 8.49 x 106m • or

  33. Challenging Addition Cont. • 8.9 x 105m + 7.6 x 103km • = 0.0089 x 105km + 7.6 x 103km • = 0.89 x 103km + 7.6 x 103km • = 8.49 x 103km • 8.49 x 103km = 8.49 x 106m

  34. Challenging Multiplication • 2.7 x 1010μl X 4.3 x 10-4cl • = 0.00027 x 1010cl X 4.3 x 10-4cl • = (0.00027 x 4.3) x 10 (10-4) (cl x cl) • = 0.001161 x 106cl2 • = 1.161 x 103cl2

  35. Challenging Division • 6.2 x 108kg • 4.2 x 10-5Mg • 6.2 x 108kg • = 4200 x 10-5kg • 6.2 • = 4200 x 10 (8- -5) • = 0.00147 x 1013 • = 1.47 x 1010

  36. SECTION 2.2 • Measurement Uncertainties

  37. Comparing Results Three students measure the width of a sheet of paper multiple times. #1 18.5cm→19.1cm, avg=18.8cm ∴(18.8 ± 0.3)cm #2 18.8cm→19.2cm, avg=19.0cm ∴(19.0 ± 0.2)cm #3 18.2cm→18.4cm, avg=18.3cm ∴(18.3 ± 0.1)cm Q: Are the three measurements in agreement? A: Students #1 & #2 have measurements that overlap, both have measurements between 18.8cm→19.1cm ∴ #1 and #2 are in agreement. However, student #3 does not have any overlap with #1 or #2, ∴ there is no agreement between student #3 and/or #1 & #2.

  38. Accuracy and Precission • Precision = • The degree of exactness with which a quantity is measured using a given instrument. • Q: Which student had the most precise measurement? • A: #3 18.2cm–18.4cm, all measurements are within ± 0.1cm.

  39. Generally when measuring quantities, the device that has the finest divisions on its scale yields the most precise measurement. • The precision of a measurement is ½ the smallest division of the instrument. • Q: How precise is a meter stick? • A: The smallest division on a meter stick is a millimeter(mm)  you can measure an object to within 0.5 mm.

  40. Accuracy = • How well the results of an experiment or measurement agree with an accepted standard value. • If the accepted/standard value of the sheet of paper was 19.0cm wide, which student was the most accurate, least accurate? • Most accurate = #2. • Least accurate = #3.

  41. When checking the accuracy of a measuring device use the Two-Point calibration method. • #1 Make sure the instrument reads 0 when it should. • #2 Make sure the instrument yields the correct measurement on some accepted standard.

  42. Techniques of Good Measurements • Measurements must be made carefully. • Common source of error = reading an instrument when looking at it from an angle  read the instrument from directly above. • Parallax = the apparent shift in position of an object when viewed from different angles.

  43. Significant digits • Significant Digits = the valid digits in a measurement. • The last (estimated) digit is called the uncertain digit. • All non zero digits in a measurement are significant.

  44. A = 1.24m B = 0.23cm • How many significant digits for A & B? • A = 3 B = 2 • Which is a more precise measurement? • A is to the nearest cm • B is to the nearest 1/100cm •  B is the more precise measurement

  45. ZEROS • Q: Are all zeros significant? • A: No • Q: Which zeros are significant? • 0.0860m # of significant digits =? • A: 3 significant digits, first 2 zeros only show the decimal place, the last one is significant, it indicates the degree of precision of the measuring device.

  46. 186 000 m • Q: How many significant digits? • A: ???????????? Cannot tell, it is ambiguous, you do not know what instrument was used to achieve this measurement, possibly 3, 4, 5 or 6 significant digits.

  47. To avoid confusion rewrite # • 186 km = 3 sig dig • 186.000 km = 6 sig dig • 186.0 km = 4 sig dig • 1.86 x 105 m = 3 sig dig • 1.86000 x 105 m = 6 sig dig • 0.186 Mm = 3 sig dig • 0.000186 Gm = 3 sig dig

  48. Rules to Determine # of Sig Dig • 1. Nonzeros are always significant. • 2. All final zeros after the decimal point are significant. • 3. Zeros between two other sig dig are always significant. • 4. Zeros used solely as placeholders are not significant.

  49. EXAMPLES # Sig Dig • 450 026 = • 6 • 0.123 = • 3 • 100 258 = • 6 • 0.000 009 = • 1 • 0.000 090 = • 2

  50. Addition & Subtraction • Perform the operation, then round off the least precise value involved. • 64.0324 • 9.641 • + 129 458.1 • = 129 531.7734 • 129 458.1 is the least precise value •  round off to 129 531.8 ,one digit past the decimal point.

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