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Measurements and Calculations Chapter 5 Measurements and Calculations

5.1 Scientific Notation Scientific nation is a method for making very large or very small numbers core compact and easier to write. Copyright © Houghton Mifflin Company

Scientific notation simply expresses a number as a product of a number between 1 and 10 and the appropriate power of 10. e.g. 93,000,000 = 9.3 X 107 *whenever the decimal point is moved to the left, the exponent of 10 is positive. Copyright © Houghton Mifflin Company

Using scientific notation with numbers smaller than 1: 0.010 = 1.0 x 10-2 *whenever the exponent is moved to the right the exponent is negative. Copyright © Houghton Mifflin Company

Practice Examples: 5.1 & 5.2 pg. 115/116 Copyright © Houghton Mifflin Company

5.2 Units The units part of a measurement tells us what scale or standard is being used to represent the results of the measurement. International System (SI): The Si units are based on the metric system and units derived from the metric system Copyright © Houghton Mifflin Company

Some Fundamental SI Units Table 5.1 Pg. 117 Physical Quantity Name of the Unit Abbreviation Mass Kilogram Kg Length Meter M Time Second s Temperature Kelvin K Copyright © Houghton Mifflin Company

Table 5.2 Copyright © Houghton Mifflin Company

Daily Planner Review 5.1 – 5.2 Chapter questions – to be collected 1-2 3 – 10 (black) Copyright © Houghton Mifflin Company

5.3 Measurements of Length, Volume, and Mass SI Unit for length = Meter 1 meter = 39.37 inches The metric system is set up in units of ten. The metric system can be expressed by powers of 10 FYI 1 inch = 2.54 centimeters Copyright © Houghton Mifflin Company

Table 5.3 Copyright © Houghton Mifflin Company

Figure 5.1: Comparison of English and metric units. Copyright © Houghton Mifflin Company

Volume Volume is the amount of three-dimensional shape occupied by a substance. SI Unit of Volume = cube – measured 1 meter on each side (13) Copyright © Houghton Mifflin Company

Figure 5.2: Cube representations. 1m X 1m X 1m = 1m3 Or 1 cubic meter 1dm3 = 1L 1 cm3 = 1 mL Copyright © Houghton Mifflin Company

Figure 5.3: A 100 mL graduated cylinder. Note the measurements on this graduated cylinder Copyright © Houghton Mifflin Company

Mass Mass is the quantity of matter present in an object. SI unit of mass = kilogram We determine mass using a balance Copyright © Houghton Mifflin Company

Table 5.5 The most commonly used metric units for mass SYMBOL GRAM EQUIVALENT KILOGAM kg 1000g = 10g 3 =1 kg GRAM g 1 g MILLIGRAM mg 0.001g = 10-3 g = 1 mg

Sections 5.1 – 5.3 1. What is the difference between a qualitative observation and a quantitative observation? 2. When you are writing numbers using scientific notation, very large numbers have ________exponents and very small numbers have ______exponents. Copyright © Houghton Mifflin Company

3. Change the following to scientific notation: 8,475,000 0.0000754 1000 0.35724 4. Why is a unit a necessary part of a measurement Copyright © Houghton Mifflin Company

5.4 Uncertainty in Measurements When we measure, there is always a level of interpretation, requiring some form of an estimate: Every measurement has some degree of uncertainty Copyright © Houghton Mifflin Company

Figure 5.5: Measuring a pin. a)The pin appears to be between 2.8 & 2.9 b) We imagine the distance between 2.8 – 2. 9 in units of 10 and estimate our answer

Certainty and uncertainty Person Possible interpretation 1 2.85 cm 2 2.84 cm 3 2.86 cm 4 2.85cm 5 Note: The first two numbers are the same they are the CERTAIN measurement. The last number , the estimate, is the UNCERTAIN measurement. When we measure always record the certain number + the first uncertain number. Copyright © Houghton Mifflin Company

Daily Planner Chapter questions pg. 149 (to be collected) Sections 5.3 and 5.4, # 13, 14, 15, 17, 19, 20 & 21 Copyright © Houghton Mifflin Company

5.5 Significant Figure The numbers recorded in a measurement (all of the certain numbers plus the first uncertain number) are termed SIGNIFICANT FIGURES (SIG FIGS) The number of sig figs is determine by he measuring device e.g. 1.86 kg = 1.86 +/- .01kg Copyright © Houghton Mifflin Company

Significant Figures Rules for counting Significant Figures 1. Nonzero Integers ALWAYS counts as significant figures. 2. Zeros – there are three classes of zeros a) Leading Zeros = Zeros that precede all of the nonzero 0.000025 They never count as significant The example above only has two sig figs = 2 & 5 Copyright © Houghton Mifflin Company

Zeros – the three classes cont., b). Captive zeros are zeros that fall between nonzero digits. They ALWAYS count as sig figs. 1.008 has four sig. figs c). Trailing Zeros are zeros at the right end of a number. They are significant only if the number is written with a decimal point. 100 = one sig. fig. 100. = three sig. fig. Copyright © Houghton Mifflin Company

3. Exact Numbers – numbers that are counted: 3 apples, 2 molecules Exact numbers are assumed to have an unlimited amount of sig. figs. 1 inch = 2.54 cm Copyright © Houghton Mifflin Company

Numbers Written in Scientific Notation 100 = 1.00 X 102 = 3 sig. figs. 0.000060 = 6.0 X 10-5 = 2 sig figs. Example 5.3 pg 125 A sample of orange juice containing 0.0108 g of vitamin C: 1.08 X 10-2 = 3 sig. figs Self Check Exercise 5.2 Copyright © Houghton Mifflin Company

Rounding Off Numbers If the digit to be removed Is less than five, the preceding digit stays the same. 1.33 = 1.3 Is ≥ 5, then the preceding digit is increased by one: 1.36 = 1.4 In a series of calculations carry the extra digits through to the final result then round off. Copyright © Houghton Mifflin Company

Rounding significant figures 4.348 If only two numbers are significant than we look at only the first number to the right of the significant figures. Copyright © Houghton Mifflin Company

Determining Sig. Figs. In Calculations Multiplication and division – the number of sig figs in the result is the same as the smallest number of sig figs. in the measurement. 4.56 X 1.4 = 6.384 → 6.4 Note: for multiplication sig. figs are counted Copyright © Houghton Mifflin Company

Addition and Subtraction – limiting term is the one with the smallest number of decimal places 12.11 For +/- decimal 18.0 places are counted 1.013 31.123 → 31.1 Copyright © Houghton Mifflin Company

Practice Example 5.4 & 5.5 pg. 128 - 129 Self Check 5.3 pg. 129 Great website for practice http://www.fordhamprep.org/gcurran/sho/sho/lessons/lesson23.htm Copyright © Houghton Mifflin Company

Focus Question 5.4 – 5.5 Why are all measurements uncertain to some extent? Mark the zeros that are sig figs: 0.003042 5. 50.0 1.4030 6. 10.47020 1000 7. 250 0.060 How many sig fig. are in each example? Copyright © Houghton Mifflin Company

Without doing the calculations, how many sig figs should be in each of the following results? 1. 3.2 + 4.17 + 1.243 2. 1.3478 – 0.02 3. 4.6 x 3.435 4. (4.2 X 10-5) X 3.74 ÷ 6.783 5. 50 – 0.00473 Copyright © Houghton Mifflin Company

5.6 Problem Solving and Dimensional Analysis How do we convert one unit of measurement to another? Conversion Factor: a ration of the two parts of the statement that relates the two units. Copyright © Houghton Mifflin Company

Converting from one unit to another Need to know an equivalence statement (12 inched = 1 ft; 2.54 cm = 1 inch) Arrange your equation so the units cancel out. Leaving you with the unit you are trying to solve for. Multiply Check that you have the right number of sig figs. Does your answer make sense? Copyright © Houghton Mifflin Company

How to use Conversion Factors Unit 1 x Conversion factor = Unit 2 Example: 2.85 cm = _____inches 2.54 cm = 1 inch 2.85 cm X 1 inch = 1.12 inched 2.54cm Copyright © Houghton Mifflin Company

Practice Pg. 132 – 134 Examples 5.6, 5.7 Self check 5.4, 5.5 Copyright © Houghton Mifflin Company

Daily Planner Homework: to be collected next class Chapter questions pg 150 #35 – 37; 39 – 43 (black). Significant Figures Worksheet Copyright © Houghton Mifflin Company

5.7 Temperature Conversion: An Approach to Problem Solving There are three scales to measure temperature: Fahrenheit Scale (F) Celsius Scale (C) Kelvin Scale or absolute scale (K) Copyright © Houghton Mifflin Company

Figure 5.6: The three major temperature scales. Note: zero points are different. The number of degree units are different. Copyright © Houghton Mifflin Company

Figure 5.7: Converting 70 degrees Celsius to Kelvin units. The size of each degree unit on the Kelvin and Celsius scale are the same. The zero points are different Copyright © Houghton Mifflin Company

Figure 5.8: Comparison of the Celsius and Fahrenheit scales. Fahrenheit degrees are smaller than Celsius degrees. Copyright © Houghton Mifflin Company

Concerting between the Kelvin and Celsius Scale 100° C + 273 = 373K 0° C + 273 = 273K -18° C +273 = 255K In each scenario above We needed to add 273 . Copyright © Houghton Mifflin Company

Formuls T°C + 273 = TK TK - 273 = T°C TK - T°C = 273 Examples 5.8, 5.9 and self check 5.6 pg 136 - 138 Copyright © Houghton Mifflin Company

Converting Between Fahrenheit and Celsius T°F = 1.80 (T°C) + 32 Examples 5.10, 5.11 5.12 and self check 5.7 & 5.8 pgs.140 - 141 Copyright © Houghton Mifflin Company

Temperature Conversion Formulas Celsius to Kelvin T°C + 273 = TK Celsius to Fahrenheit T°F = 1.80 (T°C) + 32 Kelvin to Celsius TK - 273 = T°C Fahrenheit to Celsius T°C = (T°F – 32) /1.80 Copyright © Houghton Mifflin Company

5.8 Density Density is the amount of matter present in a given volume of a substance. Density = Mass Volume Copyright © Houghton Mifflin Company

The volume of a solid object is often determined indirectly by submerging it in water and measuring the volume of water displaced. Copyright © Houghton Mifflin Company

Figure 5.9: Tank of water. Copyright © Houghton Mifflin Company

Figure 5.9: Person submerged in the tank. Copyright © Houghton Mifflin Company

Focus Questions 5.6 – 5.8 What is an equivalence statement? An equation that shows two measurements stand for the same thing How many conversion factors can be created from one equivalence statement? 2 Copyright © Houghton Mifflin Company

3. How can you decide which conversion factor to choose in a problem 3. How can you decide which conversion factor to choose in a problem? the proper conversion factor cancels out the units to be discarded leaving you with the desired unit as your result. 4. Write the conversion factors for these problems. 37 cm X ___________= ? in 4.2 qts X __________ -= ? L 2.2 kg X ___________ = ? Pounds 3.5 ml X ___________= ?m Copyright © Houghton Mifflin Company

When you make a temperature conversion, how many significant figures should be in the converted temperature? The conversion will be based upon the degree of precision of the instrument used. Copyright © Houghton Mifflin Company

Practice Problems Example 5.14; 5.15 and self check 5.9 pgs. 144- 145 Copyright © Houghton Mifflin Company

Daily Planner Homework : to be collected next class Chapter problems: pg. 150 – 151 44 – 46; 47 – 50 (blk); 51 – 55; 58, 61 Copyright © Houghton Mifflin Company