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##### Chapter 2 - Measurement

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**Chapter 2 - Measurement**• Mass and Weight • Measurement • Sig. Figs. And Rounding • Scientific Notation • Sig. Figs. in Calculations • Metric system • Measurement – Length, Mass, Volume, and Temperature • Density • Problem Solving**Mass and Weight**• Mass: The amount of matter in that body • Measured on a balance with a comparison with known masses • Weight: The measure of the earth’s gravitational attraction for that object • Measured on a scale, which measures force against a spring – Varies with location of the object • Matter:Anything that has mass and occupies space.**Nature of Measurement**• Experiments are performed. • Numerical values or data are obtained from these measurements.**Nature of Measurement**• Measurement - quantitative observation consisting of 2 parts • Part 1 - number • Part 2 - scale (unit) • Examples: • 20 grams • 6.63 Joule seconds**numerical value**70.0 kilograms = 154 pounds unit Nature of Measurement**Precision and Accuracy**• Accuracy refers to the agreement of a particular value with the truevalue. • Precisionrefers to the degree of agreement among several elements of the same quantity.**Types of Error**• Random Error (Indeterminate Error) - measurement has an equal probability of being high or low. • Systematic Error (Determinate Error) - Occurs in the same directioneach time (high or low), often resulting from poor technique.**Measurement and Significant Figures**• Numbers obtained from a measurement are never exact values. • Degree of uncertainty • Due to limitations of instrument • Skill of the individual • Recorded value should indicate uncertainty • Maximum precision • Contain all known digits • Plus one digit that is estimated • These digits are know as Significant Figures**Uncertainty in Measurement**• A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty.**Measurement and Significant Figures**Between 60 and 70 mm Estimate next digit, 4 Measurement 64 mm Between 64 and 65 mm Estimate next digit, 2 Measurement 64.2 mm Between 64.2 and 64.3 mm Estimate next digit, 5 Measurement 64.25 mm**The temperature 21.2oC is expressed to 3 significant**figures. Temperature is estimated to be 21.2oC. The last 2 is uncertain.**The temperature 22.0oC is expressed to 3 significant**figures. Temperature is estimated to be 22.0oC. The last 0 is uncertain.**The temperature 22.11oC is expressed to 4 significant**figures. Temperature is estimated to be 22.11oC. The last 1 is uncertain.**Rules for Counting Significant Figures - Zero**• 1. Nonzero integers • 2. Zeros • leading zeros • captive zeros • trailing zeros • 3. Exact numbers**Rules for Counting Significant Figures - Details**• Nonzero integersalways count as significant figures. • 3456 has • 4 sig figs.**Rules for Counting Significant Figures - Details**• Zeros • Leading zeros do not count as • significant figures. • 0.0486 has • 3 sig figs.**Rules for Counting Significant Figures - Details**• Zeros • Captive zeros always count as • significant figures. • 16.07 has • 4 sig figs.**Rules for Counting Significant Figures - Details**• Zeros • Trailing zeros are significant only • if the number contains a decimal point. • 9.300 has • 4 sig figs.**Significant Figures**All nonzero numbers are significant. 3 Significant Figures 461**Significant Figures**A zero is significant when it is between nonzero digits. 3 Significant Figures 401**Significant Figures**A zero is significant when it is between nonzero digits. 5 Significant Figures 9 3 . 0 0 6**Significant Figures**A zero is significant when it is between nonzero digits. 3 Significant Figures 9 . 0 3**Significant Figures**A zero is significant at the end of a number that includes a decimal point. 5 Significant Figures 5 5 . 0 0 0**Significant Figures**A zero is significant at the end of a number that includes a decimal point. 5 Significant Figures 2 . 1 9 3 0**Significant Figures**A zero is not significant when it is before the first nonzero digit. 1 Significant Figure 0 . 0 0 6**Significant Figures**A zero is not significant when it is before the first nonzero digit. 3 Significant Figures 0 . 7 0 9**Significant Figures**A zero is not significant when it is at the end of a number without a decimal point. 1 Significant Figure 5 0 0 0 0**Significant Figures**A zero is not significant when it is at the end of a number without a decimal point. 4 Significant Figures 6 8 7 1 0**Rules for Counting Significant Figures - Details**• Exact numbershave an infinite number of significant figures. • 1 inch = 2.54 cm, exactly**Exact Numbers**• Exact numbers have an infinite number of significant figures. • Exact numbers occur in simple counting operations 1 2 3 4 5 • Defined numbers are exact. 12 inches = 1 foot 100 centimeters = 1 meter**Rounding Off Numbers**• Often when calculations are performed extra digits are present in the results. • It is necessary to drop these extra digits so as to express the answer to the correct number of significant figures. • When digits are dropped the value of the last digit retained is determined by a process known as rounding off numbers.**Rounding off Numbers**• Rule 1. The first digit after those to be retained is 4 or less, all other digits are dropped. • 34.642 = 34.64 • Rule 2. The first digit after those to be retained is 5 or more, all other digits are dropped and the last digit is increased by one. • 34.6426 = 34.643**Rounding Off Numbers**Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed. 4 or less 80.873**Rounding Off Numbers**Rule 1. When the first digit after those you want to retain is 4 or less, that digit and all others to its right are dropped. The last digit retained is not changed. 4 or less 1.875377**5 or greater**drop these figures Rounding Off Numbers Rule 2. When the first digit after those you want to retain is 5 or greater, that digit and all others to its right are dropped. The last digit retained is increased by 1. increase by 1 5.459672 6**Scientific Notation**• Why? A convenient way of writing very small or very big numbers. • Earth age is 4,500,000,000 years • Estimated value to the nearest 0.1 billion years • Thus can be written as 4.5 x109 years. • Radius of hydrogen 0.000,000,000,037 meters • Written 3.7 x 10-11 meters**Scientific Notation**• Write a number as a power of 10 • Move the decimal point in the original number so that it is located after the first nonzero digit. • Follow the new number by a multiplication sign and 10 with an exponent (power). • The exponent is equal to the number of places that the decimal point was shifted.**Write 6419 in scientific notation.**decimal after first nonzero digit power of 10 6.419 x 103 64.19x102 641.9x101 6419. 6419**Write 0.000654 in scientific notation.**decimal after first nonzero digit power of 10 6.54 x 10-4 0.000654 0.00654 x 10-1 0.0654 x 10-2 0.654 x 10-3**Rules for Significant Figures in Mathematical Operations**• The results of a calculation cannot be more precise than the least precise measurement.**Rules for Significant Figures in Mathematical Operations**• Multiplication and Division: # sig figs in the result equals the number in the least precise measurement used in the calculation. • 6.38 2.0 = • 12.76 13 (2 sig figs)**Drop these three digits.**2.3 has two significant figures. (190.6)(2.3) = 438.38 190.6 has four significant figures. Answer given by calculator. The answer should have two significant figures because 2.3 is the number with the fewest significant figures. Round off this digit to four. 438.38 The correct answer is 440 or 4.4 x 102**Rules for Significant Figures in Mathematical Operations**• Addition and Subtraction: # sig figs in the result equals the number of decimal places in the least precise measurement. • 6.8 + 11.934 = • 18.743 18.7 (3 sig figs) • 11.934 – 10.8 = • 1.134 1.1 (2 sig figs)**125.17**129. 52.2 Add 125.17, 129 and 52.2 Least precise number. Answer given by calculator. 306.37 Round off to the nearest unit. Correct answer. 306.37**0.018286814**Drop these 6 digits. Correct answer. Answer given by calculator. Two significant figures. The answer should have two significant figures because 0.019 is the number with the fewest significant figures.**Practice**12.62 + 1.5 + 0.25 = (2.25 x 103) (4.80 x 104) = = 14.4 1.08 x 108 14.37 = 2.0 x 102 195.97 0.0007177852 = 7.18 x 10-4 10.4005625 = 10.4 10.4 + 3.75(1.5 x 10-4) =**International System(le Système International)**• Based on metric system and units derived from metric system.**International System – SI – Standard Units**Quantity Name of unit Abbreviation Length Meter m Mass Kilogram kg Temperature Kelvin K Time Second s Amount of Substance mole mol Current ampere A Luminous candela cd