Measurements & Calculations Chapter 2 All rights reserved 1 Measurement Quantitative observation Comparison based on an accepted scale e.g. Meter stick 2 Parts Number & Unit Number is a comparison Units indicate scale Numbers are meaningless without units! All rights reserved 2 2.1 Scientific Notation Technique used to express very large or very small numbers Based on powers of 10 To compare numbers written in SN First compare exponents of 10 10 9 > 10 3 10 4 > 10-4 Then compare numbers 3.6 x 10 3 > 1.2 x 10 3 Arrange the following in order from smallest to largest: 1.0 x 10 3 3.2 x 10-3 9.2 x 10-3 9.8 x 10 2 All rights reserved 3
2.1 Scientific Notation Converting to Scientific Notation: 123 -> 1.23 x 10 2 1) Move the decimal point to the right of the non-zero digit in the largest place The new number is now between 1 & 10 2) Multiply the new number by 10 n n is the number of places you moved the decimal pt 3) Determine the sign on the exponent n If the decimal pt was moved left, n is + If the decimal pt was moved right, n is If the decimal pt was not moved, n is 0 All rights reserved 4 2.1 Scientific Notation Converting to Standard Form 1.23 x 10 2 -> 123 1) Determine the sign of n of 10 n If n is + the decimal pt will move to the right If n is the decimal pt will move to the left 2) Determine the value of the exponent of 10 Is the number of places to move the decimal pt 3) Move the decimal pt and rewrite the number All rights reserved 5 2.1 Scientific Notation Convert the following 1) From SN to standard form 1.35 x 10-6 4.592 x 10 1 8.1 x 10 3 2) From standard form to SN 1,000,310 0.00025 7.31 All rights reserved 6
2.2 Units All units in the metric system are related to the fundamental unit by a power of 10 The power of 10 is indicated by a prefix The prefixes are always the same, regardless of the fundamental unit All rights reserved 7 2.2 Units Table 2.2 Commonly Used Prefixes Prefix Symbol Value mega M 1,000,000 kilo k 1,000 deci d 0.1 centi c 0.01 milli m 0.001 micro μ 0.000001 nano n 0.000000001 SN 10 6 10 3 10-1 10-2 10-3 10-6 10-9 All rights reserved 8 2.3 Length, Volume & Mass Length SI unit = meter (m) About 3½ inches longer than a yard Commonly use centimeters (cm) 1 cm = 10-2 m = 0.01 m = 10 mm 1 inch = 2.54 cm (exact number) All rights reserved 9
2.3 Length, Volume & Mass Volume - amount of three-dimensional space SI unit = m 3 Commonly measure solid volume in cm 3 1 cm 3 = (10-2 m) 3 = 10-6 m 3 = 0.000001 m 3 Commonly measure liquid or gas volume in ml 1 L is slightly larger than 1 quart 1 ml = 10-3 L = 0.001 L 1 ml = 1 cm 3 All rights reserved 10 2.3 Length, Volume & Mass Mass - amount of matter present SI unit = kg Commonly measure mass in g or mg 1 kg = 2.2046 pounds, 1 lbs = 453.59 g 1 kg = 10 3 g = 1000 g 1 mg = 10-3 g = 0.001 g All rights reserved 11 2.4 Uncertainty in Measurement A measurement always has some amount of uncertainty Uncertainty comes from limitations of the techniques used for comparison To understand how reliable a measurement is, we need to understand the limitations of the measurement All rights reserved 12
Significant figures (SF S) are used to indicate the uncertainty of a measurement Measurements are reported with one uncertain digit Unless stated otherwise, the uncertainty in the last digit is ±1 210.3 ml is accepted to mean 210.2, 210.3 or 210.4 ml 210.3 ± 0.1 ml 0.57 cm is accepted to mean 0.56, 0.57 or 0.58 cm 0.57 ± 0.01 cm All of the digits of a measurement, are said to be SF s All rights reserved 13 Rules for Counting SF s Nonzero integers of measurements are always significant Zeros Leading zeros never count as SF s Captive zeros are always significant Trailing zeros are significant if there is a decimal pt Exact numbers have an unlimited number of SF s 2.54 cm/in 1000 ml in a L 121 students All rights reserved 14 Indicate the number of SF s in each of the following values: 32410 45011 9.00300 730,000 0.0130520 100100.00 All rights reserved 15
Rules for Rounding Off If the digit to be removed is less than 5, the preceding digit stays the same is greater than 5, the preceding digit is increased by 1 is equal to 5, the preceding digit is rounded such to keep it even 34.5 rounded to 2 SF s is 34 6575 rounded to 3 SF s is 6580 Generally, this is a calculated value, in which case, if you carry a few more non-sf s, you ll find it isn t exactly 5 and one of the first two rules can be applied All rights reserved 16 Round 321.2565 to the indicated SF s 2 SF s 3 SF s 4 SF s 5 SF s 6 SF s All rights reserved 17 Exact numbers Numbers known with certainty Unlimited number of SF s Counting numbers number of sides on a square number of people in a room Defined numbers 1 minute = 60 seconds 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm 1 kg = 1000 g, 1 LB = 16 oz 1000 ml = 1 L; 1 gal = 4 qts. All rights reserved 18
Calculating with SF s Calculators/computers do not know about SF s!!! Exact numbers do not affect the number of SF s in an answer Answers to calculations must be rounded to the proper number of SF s Carry 2 non-sf s thru calculations Use vertical dashed line between SF s and non-sf s Round at the end of the calculation All rights reserved 19 Multiplying & Dividing with SF s Result has the same number of SF s as the measurement with the smallest number of SF s Count the number of SF s in each measurement Round the result so it has the same number of SF s as the measurement with the smallest number of SF s ( 4.5 cm) ( 0.200 cm) = 0.90M0 cm 2 = 0.90 cm 2 All rights reserved 20 Perform the following calculations, give an intermediate answer with 2 non-sf s and a dashed line, and give a final answer to the appropriate SF s 4.312 x 386.51 (2.3 x 1.45)/3 All rights reserved 21
Adding & Subtracting with SF s Result is limited by the uncertainty provided by each measurement, but the answer is reported with only one uncertain digit. Find last SF in each measurement Find which one is left-most Round answer to the same uncertainty, using zeroes as decimal place holders 4510 ml + 265 ml 4775 ml All rights reserved 22 2.6 Problem Solving & Dimensional Analysis Conversion factors are relationships between two units May be exact or measured Conversion factors are generated from equivalence statements 1 inch = 2.54 cm Two conversion factors are generated from each equivalence statement 1 inch 2.54 cm and 2.54 cm 1 inch All rights reserved 23 2.6 Problem Solving & Dimensional Analysis Chose the conversion factor that cancels unwanted units To convert inches to cm use: 2.54 cm 1 inch To convert cm to inches use: 1 inch 2.54 cm May string conversion factors To convert inches to km use: All rights reserved 24
2.6 Problem Solving & Dimensional Analysis Converting One Unit to Another: Convert 342 kg to ounces Find the relationship(s) between the starting and goal units. Write an equivalence statement for each relationship. Kg -> g -> lbs -> oz Kg = 10 3 g, 453.6 g = 1 lb, 1 lb = 16 oz Write conversion factors for each eq. statement. kg 10 3 g or 103 g kg 453.6 g 1 lb or 1 lb 453.6 g 1 lb 16 oz or 16 oz 1 lb All rights reserved 25 2.6 Problem Solving & Dimensional Analysis Convert 342 kg to ounces Arrange the conversion factor(s) to cancel starting unit and result in goal unit. Ê ( 342 kg) 103 gˆ Ê lb ˆ Ê 16 oz ˆ Á Á Á Ë kg Ë 453.6 g Ë lb Check that the units cancel properly Perform the calculation to give the answer with the proper unit and SF s = 12,0M63 oz = 12,100 oz Check that your answer makes sense! All rights reserved 26 2.7 Temperature Conversions K & C conversions: Same size degree Different 0 degree T K = T C + 273.15 or T C = T K - 273.15 All rights reserved 27
2.7 Temperature Conversions F & C conversions: Different size degree Different O degree T F = 1.80(T C ) + 32.00 or T C = (T F - 32.00)/1.80 All rights reserved 28 2.7 Temperature Conversions F & K conversions: Combine two temperature conversions All rights reserved 29 2.7 Temperature Conversions Report converted temperature to the same uncertainty as the original temp. Convert the 74.0 F to K All rights reserved 30
2.8 Density Density is a property of matter representing the mass per unit volume Solids = g/cm 3 Liquids = g/ml Gases = g/l Density = Mass Volume Density : solids > liquids >>> gases In a heterogeneous mixture, denser object sinks All rights reserved 31 2.8 Density Density is a conversion factor between mass & volume. Density = Mass Volume Calculate the volume in L that would be occupied by a 1.00 lb brick of gold. The density of gold is 19.32 g/cm 3 All rights reserved 32