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Measurement

Measurement. How far, how much, how many?. PROBLEM SOLVING. STEP 1: Understand the Problem STEP 2: Devise a Plan STEP 3: Carry Out the Plan STEP 4: Look Back. Step 1. Understand the Problem. Step 2. Devise a Plan. Step 3. Carry Out the Plan. Step 4. Look Back. A Number A Quantity

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Measurement

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

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

  3. Step 1. Understand the Problem

  4. Step 2. Devise a Plan

  5. Step 3. Carry Out the Plan

  6. Step 4. Look Back

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

  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

  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

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

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

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

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

  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

  15. Let’s use a golf analogy

  16. Accurate? No Precise? Yes

  17. Accurate? Yes Precise? Yes

  18. Precise? No Accurate? Maybe?

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

  20. Accuracy vs. Precision Synonyms for Accuracy… Truevalue Correct Bulls eye! SingleMeasurement

  21. Accuracy vs. Precision Closely Grouped Multiple Measurements Synonymsfor precision… Repeatable

  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!

  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

  24. SCALES 0 1 0.2, 0.3, 0.4?

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

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

  27. Certain Digits Uncertain Digit 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

  28. Significant figures: Rules for zeros • Leading zeros are notsignificant. 0.00421 - three significant figures 4.21 x 10-3 Leading zero Notice zeros are not written in scientific notation Captive zeros are significant. 4012 - four significant figures 4.012 x 103 Captive zero Notice zero is written in scientific notation

  29. Significant figures: Rules for zeros Trailing zeros before the decimal are notsignificant. 4210000 - three significant figures Trailing zero Trailing zeros after the decimal are significant. 114.20 - five significant figures Trailing zero

  30. 123 grams 1005 mg 250 kg 250.0 kg 2.50 x 102 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 How Many Significant figures?

  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 1012 • or • $3,000,000,000,000.00

  32. Significant figures:Rules for zeros • 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. 0.003210 = 3.210 x 10-3 Four Significant Figures

  33. A comparison of masses Compare the mass of a block of wood that was taken on 4 different balances.

  34. Average mass calculation

  35. 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?

  36. your value accepted value Percent Error • Indicates accuracy of a measurement

  37. % error = 2.94 % 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.

  38. 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

  39. Scientific notation • Method to express really big or small numbers. • Format is Mantissa x Base Power Decimal part of original number Decimals you moved We just move the decimal point around.

  40. 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 10X. • 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 108

  41. 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

  42. Scientific notation • Most scientific calculators use scientific notation when the numbers get very large or small. • How scientific notation is • displayed can vary. • It may use x10n • or may be displayed • using an E or e. • They usually have an Exp or EE • button. This is to enter in the exponent. 1.44939 E-2

  43. Examples • 378 000 • 3.78 x 10 5 • 8931.5 • 8.9315 x 103 • 0.000 593 • 5.93 x 10 - 4 • 0.000 000 40 • 4.0 x 10 - 7

  44. Expand • 1 x 104 • 10,000 • 5.60 x 1011 • 560,000,000,000 • 1 x 10-5 • 0.000 01 • 5.02 x 10-8 • 0.000 000 0502

  45. 123.45987 g + 234.11 g 357.56987 g 357.57 g • 805.4 g • 721.67912 g • 83.72088 g • 83.7 g 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.

  46. 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/m3 • = 2.8 kg / m3 • (2.1 m - only has two significant figures)

  47. Significant figuresand 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? 0.474137931 • speed = 11 km / 23.2 min • = 0.47 km / min

  48. 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 m3 • = 91 m3 2.1 m 2.351 m 18.5 m

  49. Scientific Notation (Multiplication) (3.0 x 104) x (3.0 x 105) = 9.0 x 109 (6.0 x 105) x (2.0 x 104) = 12 x 109 But 12 x 109 = 1.2 x 1010

  50. Scientific Notation (Division) 2.0 x 106 = 1.0 x 104 1.0 x 104 = 2.0 x 106 2.0 x 102 0.50 x 10-2 = 5.0 x 10-3

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