2. 2 - 1 Measurement How far, how much, how many?

3. 2 - 2 PROBLEM SOLVING STEP 1: Understand the Problem STEP 2: Devise a Plan STEP 3: Carry Out the Plan STEP 4: Look Back

4. 2 - 3 Step 1. Understand the Problem

5. 2 - 4 Step 2. Devise a Plan

6. 2 - 5 Step 3. Carry Out the Plan

7. 2 - 6 Step 4. Look Back

8. 2 - 7 A Measurement A Number A Quantity An implied precision 15 1000000000 0.00056 A Unit A meaning pound Liter Gram Hour degree Celsius

9. 2 - 8 Implied versus Exact An implied or measured quantity has significant figures associated with the measurement 1 mile = 1603 meters Exact - defined measured - 4 sig figs An exact number is not measured, it is defined or counted; therefore, it does not have significant figures or it has an unlimited number of significant figures. 1 kg = 1000 grams 1.0000000 kg = 1000.0000000 grams

10. 2 - 9 Types of measurement Quantitative- use numbers to describe measurement– test equipment, counts, etc. Qualitative- use descriptions without numbers to descript measurement- use five senses to describe 4 feet extra large Hot 100ºF

11. 2 - 10 Scientists Prefer Quantitative- easy check Easy to agree upon, no personal bias The measuring instrument limits how good the measurement is

12. 2 - 11 Uncertainty in Measurement All measurements contain some uncertainty. We make errors Tools have limits Uncertainty is measured with Accuracy AccuracyHow close to the true value Precision PrecisionHow close to each other

13. 2 - 12 Accuracy Measures how close the experimental measurement is to the accepted, true or book value for that measurement

14. 2 - 13 Precision Is the description of how good that measurement is, how many significant figures it has and how repeatable the measurement is.

15. 2 - 14 Differences Accuracy can be true of an individual measurement or the average of several Precision requires several measurements before anything can be said about it

16. 2 - 15 Let’s use a golf analogy

17. 2 - 16 Accurate?No Precise?Yes

18. 2 - 17 Accurate?Yes Precise?Yes

19. 2 - 18 Precise?No Accurate?Maybe?

20. 2 - 19 Accurate?Yes Precise?We can’t say!

21. 2 - 20 Accuracy vs. Precision Correct True value Single Measurement Bulls eye! Synonyms for Accuracy…

22. 2 - 21 Accuracy vs. Precision Synonyms for precision… Closely Grouped Repeatable Multiple Measurements

23. 2 - 22 Significant figures The number of significant digits is independent of the decimal point. 25500 2550 255 25.5 2.55 0.255 0.0255 These numbers All have three significant figures!

24. 2 - 23 Significant Figures Imply how the quantity is measured and to what precision. Are always dependant upon the equipment or scale used when making the measurement

25. 2 - 24 SCALES 0 1 0.2, 0.3, 0.4?

26. 2 - 25 SCALES 0 1 0.26, 0.27, or 0.28? 0.2, 0.3, 0.4?

27. 2 - 26 SCALES 0 1 0.26, 0.27, or 0.28? 0 1 0.262, 0.263, 0.264? 0.2, 0.3, 0.4? 0 1

28. 2 - 27 Significant figures Method used to express accuracy and precision. You can’t report numbers better than the method used to measure them. 67.2 units = three significant figures Certain Digits Uncertain Digit

29. 2 - 28 Significant figures: Rules for zeros are not Leading zeros are not significant. 0.00421 - three significant figures 4.21 x 10 -3 Leading zero are Captive zeros are significant. 4012 - four significant figures 4.012 x 10 3 Captive zero Notice zeros are not written in scientific notation Notice zero is written in scientific notation

30. 2 - 29 Significant figures: Rules for zeros are not Trailing zeros before the decimal are not significant. 4210000 - three significant figures Trailing zero are Trailing zeros after the decimal are significant. 114.20 - five significant figures Trailing zero

31. 2 - 30 How Many Significant figures? 123 grams 1005 mg 250 kg 250.0 kg 2.50 x 10 2 kg 0.0005 L 0.00050 L 5.00 x 10 -4 L 3 significant figures 4 significant figures 2 significant figures 4 significant figures 3 significant figures 1 significant figures 2 significant figures 3 significant figures

32. 2 - 31 Significant figures Zeros are what will give you a headache! They are used/misused all of the time.Example The press might report that the federal deficit is three trillion dollars. What did they mean? $3 x 10 12 or $3,000,000,000,000.00

33. 2 - 32 Significant figures: Rules for zeros Scientific notation Scientific notation - can be used to clearly express significant figures. A properly written number in scientific notation always has the the proper number of significant figures. 32103.210 0.003210 = 3.210 x 10 -3 Four Significant Figures Four Significant Figures

34. 2 - 33 How many significant figures? Using the measurements made of the wooden block, how many significant figures do each of the quantities measured have? 1.25 m 1.1 m 1 m

35. 2 - 34 A comparison of masses Compare the mass of a block of wood that was taken on 4 different balances.

36. 2 - 35 Average mass calculation

37. 2 - 36 Experimental Error The accuracy is measured by comparing the result of your experiment with a true or book value. The block of wood is known to weigh exactly 1.5982 grams. The average value you calculated is 1.48 g. Is this an accurate measurement?

38. 2 - 37 Percent Error Percent Error Indicates accuracy of a measurement your value accepted value

39. 2 - 38 Percent Error A student determines the density of a substance to be 1.40 g/mL. Find the % error if the accepted value of the density is 1.36 g/mL. % error = 2.94 %

40. 2 - 39 Scientific Notation Is used to write very, very small numbers or very large numbers Is used to imply a specific number of significant figures Uses exponentials or powers of 10 large positive exponentials imply numbers much greater than 1 negative exponentials imply numbers smaller than 1

41. 2 - 40 Scientific notation Method to express really big or small numbers. Format isMantissa x Base Power Decimal part of original number Decimals you moved We just move the decimal point around.

42. 2 - 41 Scientific Notation

43. 2 - 42 Scientific notation If a number is larger than 1 The original decimal point is moved X places to the left. The resulting number is multiplied by 10 X. The exponent is the number of places you moved the decimal point. The exponent is a positive value. 1 2 3 0 0 0 0 0 0 = 1.23 x 10 8

44. 2 - 43 Scientific notation If a number is smaller than 1 The original decimal point is moved X places to the right. The resulting number is multiplied by 10 -X. The exponent is the number of places you moved the decimal point. The exponent is a negative value. 0. 0 0 0 0 0 0 1 2 3 = 1.23 x 10 -7

45. 2 - 44 Most scientific calculators use scientific notation when the numbers get very large or small. How scientific notation is displayed can vary. It may use x10 n or may be displayed using an E or e. They usually have an Exp or EE button. This is to enter in the exponent. Scientific notation 1.44939 E-2

46. 2 - 45 Examples 378 000 3.78 x 10 5 8931.5 8.9315 x 10 3 0.000 593 5.93 x 10 - 4 0.000 000 40 4.0 x 10 - 7

47. 2 - 46 Expand 1 x 10 4 10,000 5.60 x 10 11 560,000,000,000 1 x 10 -5 0.000 01 5.02 x 10 -8 0.000 000 0502

48. 2 - 47 Significant figures and calculations Addition and subtraction Report your answer with the same number of digits to the right of the decimal point as the number having the fewest to start with. 123.45987 g + 234.11 g 357.56987 g 357.57 g 805.4 g - 721.67912 g 83.72088 g 83.7 g

49. 2 - 48 Addition and Subtraction In addition and subtraction, the decimal is fixed. The resulting answer will be dependant upon the least significant place holder. + - mL km

50. 2 - 49 Significant figures and calculations Multiplication and division. Report your answer with the same number of digits as the quantity have the smallest number of significant figures. Example. Density of a rectangular solid. 251.2 kg / [ (18.5 m) (2.351 m) (2.1m) ] = 2.750274 kg/m 3 = 2.8 kg / m 3 (2.1 m - only has two significant figures)

51. 2 - 50 Multiplication or Division Since multiplication and division results in a floating decimal, the number of significant figures that result from one of these processes will always be the least number of significant figures.

52. 2 - 51 Significant figures and calculations An answer can’t have more significant figures than the quantities used to produce it.Example How fast did the man run if he went 11 km in 23.2 minutes? speed = 11 km / 23.2 min = 0.47 km / min 0.474137931

53. 2 - 52 How many significant figures? What is the Volume of this box? Volume = length x width x height = (18.5 m x 2.351 m x 2.1 m) = 91.33635 m 3 = 91 m 3 18.5 m 2.351 m 2.1 m

54. 2 - 53 Scientific Notation (Multiplication) (3.0 x 10 4 ) x (3.0 x 10 5 ) = 9.0 x 10 9 (6.0 x 10 5 ) x (2.0 x 10 4 ) = 12 x 10 9 But 12 x 10 9 = 1.2 x 10 10

55. 2 - 54 Scientific Notation (Division) 2.0 x 10 6 1.0 x 10 4 = 2.0 x 10 2 1.0 x 10 4 2.0 x 10 6 = 0.50 x 10 -2 = 5.0 x 10 -3

56. 2 - 55 Scientific Notation Add & Subtract 6.4 x 10 4 (2.3 x 10 4 ) + (4.1 x 10 4 ) = *Exponent must be the same!*

57. 2 - 56 (1.400 x 10 5 ) + (3.200 x 10 3 ) = (140.0 x 10 3 ) + (3.200 x 10 3 ) = 143.2 x 10 3 = 1.432 x 10 5 Scientific Notation (+ and -)

58. 2 - 57 Rounding off numbers After calculations, the last thing you do is round the number to correct number of significant figures. If the first insignificant digit is 5 or more, - you round up If the first insignificant digit is 4 or less, - you round down.

59. 2 - 58 If a set of calculations gave you the following numbers and you knew each was supposed to have four significant figures then - 9 2.5795035 becomes 2.580 0 34.204221 becomes 34.20 Rounding off 1st insignificant digit

60. 2 - 59 Measurements Many different systems for measuring the world around us have developed over the years. People in the U.S. rely on the English System. Scientists make use of SI units so that we all are speaking the same measurement language.

61. 2 - 60 Units are important 45 has little meaning, just a number 45 g has some meaning - mass 45 g /mL more meaning - density

62. 2 - 61 Metric Units Metric Units One base unit for each type of measurement. Use a prefix to change the size of unit. Some common base units. TypeNameSymbol Massgram g Lengthmeter m Volumeliter L Timesecond s TemperatureKelvin K Energyjoule J Units

63. 2 - 62 Metric prefixes Changing the prefix alters the size of a unit. Powers of Ten http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html Prefix Symbol Factor gigaG10 9 1 000 000 000 megaM10 6 1 000 000 kilok10 3 1 000 hectoh10 2 100 decada10 1 10 base-10 0 1 decid10 -1 0.1 centic10 -2 0.01 millim10 -3 0.001 micro  or mc10 -6 0.000 001 nanon10 -9 0.000 000 001

64. 2 - 63

65. 2 - 64 Measuring mass Mass Mass - the quantity of matter in an object. Weight Weight - the effect of gravity on an object. Since the Earth’s gravity is relatively constant, we can interconvert between weight and mass. kilogram (kg) gram (g) The SI unit of mass is the kilogram (kg). However, in the lab, the gram (g) is more commonly used.

66. 2 - 65 Temperature Units of measurement Fahrenheit, Celsius, Kelvin Method of measurement

67. 2 - 66 Temperature conversion Temperature - measure of heat energy. Three common scales used Fahrenheit, Celsius and Kelvin. o F= (32 o F + o C) X o C = ( o F - 32 o F) K = ( o C + 273) X SI unit 5oC9oF5oC9oF 9oF5oC9oF5oC 1 K 1 o C

68. 2 - 67 Example. o F to K If the temperature is 75.0 o F, what is it in K? o C = (75.0 o F - 32) 5 9 = 23.9 K= 23.9 o C + 273 = 296.9 First convert to o C Then convert to K

69. 2 - 68 Measuring time The SI unit is the second (s). For longer time periods, we can use SI prefixes or units such as minutes (min), hours (h), days (day) and years. Months are never used - they vary in size.

70. 2 - 69 Derived Units QuantityDefinitionDerived Unit Arealength x lengthm 2 Volumelength x length x lengthm 3 densitymass per unit volumekg/m 3 speeddistance per unit timem/s accelerationspeed per unit timem/s 2 Forcemass x accelerationkg m/s 2 N Pressureforce per unit areakg/m s 2 Pa Energyforce x distancekg m 2 / s 2 J

71. 2 - 70 Measuring volume Volume Volume - the amount of space that an object occupies. liter (L)The base metric unit is the liter (L). milliliter (mL)The common unit used in the lab is the milliliter (mL). cm 3 & ccOne milliliter is exactly equal to one cm 3 & cc. SIm 3The derived SI unit for volume is the m 3 which is too large for convenient use.

72. 2 - 71 Density Density is an intensive property of a substance based on two extensive properties. Common units are g / cm 3 or g / mL. g / cm 3 Air 0.0013 Bone1.7 - 2.0 Water1.0Urine1.01 - 1.03 Gold19.3Gasoline0.66 - 0.69 Density = Mass Volume cm 3 = mL

73. 2 - 72 Example. Density calculation What is the density of 5.00 mL of a fluid if it has a mass of 5.23 grams? d = mass / volume d = 5.23 g / 5.00 mL d = 1.05 g / mL What would be the mass of 1.00 liters of this sample?

74. 2 - 73 Example. Density calculation What would be the mass of 1.00 liters of the fluid sample? The density was 1.05 g/mL. density = mass / volume somass = volume x density mass = 1.00 L x 1000 x 1.05 = 1.05 x 10 3 g ml L g mL

75. 2 - 74 Converting units Factor label method Regardless of conversion, keeping track of units makes thing come out right Must use conversion factors - The relationship between two units Canceling out units is a way of checking that your calculation is set up right!

76. 2 - 75 Common conversion factors Factor EnglishFactor 1 gallon = 4 quarts4 qt/gal or 1gal/4 qt 1 mile = 5280 feet5280 ft/mile or 1 mile/5280 ft 1 ton = 2000 pounds 2000 lb/ton or 1 ton/2000 lb Common English to Metric 1 liter = 1.057 quarts1.057 qt/L or 1 L/1.057 qt or 0.946 L/qt 1 kilogram = 2.2 pounds2.2 lb/kg or 1 kg/2.2 lb or 0.454 kg/lb 1 meter = 1.094 yards1.094 yd/m or 1m/1.094 yd or 0.917m/yd 1 inch = 2.54 cm2.54 cm/inch or 1 in/2.54 cm

77. 2 - 76 Example. Metric conversion How many milligrams are in a kilogram?

78. 2 - 77 Example A nerve impulse in the body can travel as fast as 400.0 feet/second. What is its speed in meters/min ? Conversions Needed 1 meter = 3.3 feet 1 minute= 60 seconds

79. 2 - 78 m 400.0 ft 1 m 60 sec min 1 sec 3.3 ft 1 min Example m 400.0 ft 1 m 60 sec min 1 sec 3.3 ft 1 min ? ? =xx ? ? =xx m min....Fast 7273