1 Chapter 2 Measurements and Calculations

2 Types of measurement l Quantitative- use numbers to describe l Qualitative- use description without numbers l 4 feet l extra large l Hot l 100ºF

3 Scientific Notation l A decimal point is in standard position if it is behind the first non-zero digit. l Let X be any number and let N be that number with the decimal point moved to standard position. Then: l If 0 < X < 1 then X = N x 10 negative number l If 1 < X < 10 then X = N x 10 0 l If X > 10 then X = N x 10 positive number

4 Some examples l becomes 8.7 x 10¯ 4 l 9.8 becomes 9.8 x 10 0 (the 10 0 is seldom written) l 23,000,000 becomes 2.3 x 10 7

5 Adding and Subtracting l All exponents MUST BE THE SAME before you can add and subtract numbers in scientific notation. l The actual addition or subtraction will take place with the numerical portion, NOT the exponent.

6 Adding and Subtracting l Example: 1.00 x x 10 2 l A good rule to follow is to express all numbers in the problem in the highest power of ten. l Convert 1.00 x 10 2 to 0.10 x 10 3, then add: 1.00 x x 10 3 = 1.10 x 10 3

7 Multiplication and Division l Multiplication: Multiply the decimal portions and add the exponential portions. l Example #1: (3.05 x 10 6 ) x (4.55 x 10¯ 10 ) l Here is the rearranged problem: (3.05 x 4.55) x ( (-10) ) l You now have 13 x 10¯ 4 = 1.3 x 10¯ 3

8 Multiplication and Division l Division: Divide the decimal portions and subtract the exponential portions. l Example: (3.05 x 10 6 ) ÷ (4.55 x 10¯ 10 ) l Here is the rearranged problem: (3.05 ÷ 4.55) x ( (-10) ) l You now have 0.7 x = 7.0 x 10 15

9 Scientists prefer l Quantitative- easy check l Easy to agree upon, no personal bias l The measuring instrument limits how good the measurement is

10 How good are the measurements? l Scientists use two word to describe how good the measurements are l Accuracy- how close the measurement is to the actual value l Precision- how well can the measurement be repeated

11 Differences l Accuracy can be true of an individual measurement or the average of several l Precision requires several measurements before anything can be said about it l examples

12 Let’s use a golf anaolgy

13 Accurate?No Precise?Yes

14 Accurate?Yes Precise?Yes

15 Precise?No Accurate?Maybe?

16 Accurate?Yes Precise?We cant say!

17 In terms of measurement l Three students measure the room to be 10.2 m, 10.3 m and 10.4 m across. l Were they precise? l Were they accurate?

18 Significant figures (sig figs) l How many numbers mean anything l When we measure something, we can (and do) always estimate between the smallest marks

19 Significant figures (sig figs) l The better marks the better we can estimate. l Scientist always understand that the last number measured is actually an estimate 21345

20 Sig Figs l What is the smallest mark on the ruler that measures cm? l 142 cm? l 140 cm? l Here there’s a problem does the zero count or not? l They needed a set of rules to decide which zeroes count. l All other numbers do count

21 Which zeros count? l Those at the end of a number before the decimal point don’t count l l If the number is smaller than one, zeroes before the first number don’t count l 0.045

22 Which zeros count? l Zeros between other sig figs do. l 1002 l zeroes at the end of a number after the decimal point do count l l If they are holding places, they don’t. l If they are measured (or estimated) they do

23 Sig Figs l Only measurements have sig figs. l Counted numbers are exact l A dozen is exactly 12 l A a piece of paper is measured 11 inches tall. l Being able to locate, and count significant figures is an important skill.

24 Sig figs. l How many sig figs in the following measurements? l 458 g l 4085 g l 4850 g l g l g l g

25 Sig Figs. l g l 4050 g l g l g l g l Next we learn the rules for calculations

26 More Sig Figs

27 Problems l 50 is only 1 significant figure l if it really has two, how can I write it? l A zero at the end only counts after the decimal place l Scientific notation l 5.0 x 10 1 l now the zero counts.

28 Adding and subtracting with sig figs l The last sig fig in a measurement is an estimate. l Your answer when you add or subtract can not be better than your worst estimate. l have to round it to the least place of the measurement in the problem

29 For example l First line up the decimal places Then do the adding Find the estimated numbers in the problem This answer must be rounded to the tenths place

30 Rounding rules l look at the number behind the one you’re rounding. l If it is 0 to 4 don’t change it l If it is 5 to 9 make it one bigger l round to four sig figs l to three sig figs l to two sig figs l to one sig fig

31 Practice l l l l 6.0 x x 10 3 l l l l 6.0 x x 10 -3

32 Multiplication and Division l Rule is simpler l Same number of sig figs in the answer as the least in the question l 3.6 x 653 l l 3.6 has 2 s.f. 653 has 3 s.f. l answer can only have 2 s.f. l 2400

33 Multiplication and Division l Same rules for division l practice l 4.5 / l 4.5 x l x.043 l / 1983 l / 714

34 The Metric System An easy way to measure

35 Measuring l The numbers are only half of a measurement l It is 10 long l 10 what. l Numbers without units are meaningless. l How many feet in a yard l A mile l A rod

36 The Metric System l Easier to use because it is a decimal system l Every conversion is by some power of 10. l A metric unit has two parts l A prefix and a base unit. l prefix tells you how many times to divide or multiply by 10.

37 Base Units l Length - meter more than a yard - m l Mass - grams - a bout a raisin - g l Time - second - s l Temperature - Kelvin or ºCelsius K or C l Energy - Joules- J l Volume - Liter - half f a two liter bottle- L l Amount of substance - mole - mol

38 SI Prefix Conversions mega-M10 6 deci-d10 -1 centi-c10 -2 milli-m10 -3 PrefixSymbolFactor micro-  nano-n10 -9 pico-p kilo-k10 3 move left move right BASE UNIT

39 Prefixes l kilo k 1000 times l deci d 1/10 l centi c 1/100 l milli m 1/1000 l kilometer - about 0.6 miles l centimeter - less than half an inch l millimeter - the width of a paper clip wire

40 Dimensional Analysis l The “Factor-Label” Method –Units, or “labels” are canceled, or “factored” out

41 Dimensional Analysis l Steps: 1. Identify starting & ending units. 2. Line up conversion factors so units cancel. 3. Multiply all top numbers & divide by each bottom number. 4. Check units & answer.

42 Dimensional Analysis l Lining up conversion factors: 1 in = 2.54 cm 2.54 cm 1 in = 2.54 cm 1 in 1 in = 1 1 =

43 Dimensional Analysis l How many milliliters are in 1.00 quart of milk? 1.00 qt1 L qt = 946 mL qtmL 1000 mL 1 L 

44 Dimensional Analysis l You have 1.5 pounds of gold. Find its volume in cm 3 if the density of gold is 19.3 g/cm 3. lbcm lb1 kg 2.2 lb = 35 cm g 1 kg 1 cm g

45 Dimensional Analysis l How many liters of water would fill a container that measures 75.0 in 3 ? 75.0 in 3 (2.54 cm) 3 (1 in) 3 = 1.23 L in 3 L 1 L 1000 cm 3

46 Dimensional Analysis 5) Your European hairdresser wants to cut your hair 8.0 cm shorter. How many inches will he be cutting off? 8.0 cm1 in 2.54 cm = 3.2 in cmin

47 Dimensional Analysis 6) Taft football needs 550 cm for a 1st down. How many yards is this? 550 cm1 in 2.54 cm = 6.0 yd cmyd 1 ft 12 in 1 yd 3 ft

48 Dimensional Analysis 7) A piece of wire is 1.3 m long. How many 1.5-cm pieces can be cut from this wire? 1.3 m100 cm 1 m = 86 pieces cmpieces 1 piece 1.5 cm

49 Volume l calculated by multiplying L x W x H l Liter the volume of a cube 1 dm (10 cm) on a side l so 1 L = 10 cm x 10 cm x 10 cm l 1 L = 1000 cm 3 l 1/1000 L = 1 cm 3 l 1 mL = 1 cm 3

50 Volume l 1 L about 1/4 of a gallon - a quart l 1 mL is about 20 drops of water or 1 sugar cube

51 Mass l weight is a force, is the amount of matter. l 1gram is defined as the mass of 1 cm 3 of water at 4 ºC. l 1000 g = 1000 cm 3 of water l 1 kg = 1 L of water

52 Mass l 1 kg = 2.5 lbs l 1 g = 1 paper clip l 1 mg = 10 grains of salt or 2 drops of water.

53 Converting khDdcm l how far you have to move on this chart, tells you how far, and which direction to move the decimal place. l The box is the base unit, meters, Liters, grams, etc.

54 Conversions l Change 5.6 m to millimeters khDdcm l starts at the base unit and move three to the right. l move the decimal point three to the right 5600

55 Conversions l convert 25 mg to grams l convert 0.45 km to mm l convert 35 mL to liters l It works because the math works, we are dividing or multiplying by 10 the correct number of times khDdcm

56 Conversions l Change 5.6 km to millimeters khDdcm

57 Measuring Temperature l Celsius scale. l water freezes at 0ºC l water boils at 100ºC l body temperature 37ºC l room temperature ºC 0ºC

58 Measuring Temperature l Kelvin starts at absolute zero (-273 º C) l degrees are the same size l C = K -273 l K = C l Kelvin is always bigger. l Kelvin can never be negative. 273 K

59 Which is heavier? it depends

60 Density l how heavy something is for its size l the ratio of mass to volume for a substance l D = M / V l Independent of how much of it you have l gold - high density l air low density.

61 Calculating l The formula tells you how l units will be g/mL or g/cm 3 l A piece of wood has a mass of 11.2 g and a volume of 23 mL what is the density? l A piece of wood has a density of 0.93 g/mL and a volume of 23 mL what is the mass?

62 Calculating l A piece of wood has a density of 0.93 g/mL and a mass of 23 g what is the volume? l The units must always work out. l Algebra 1 l Get the thing you want by itself, on the top. l What ever you do to onside, do to the other

63 Floating l Lower density floats on higher density. l Ice is less dense than water. l Most wood is less dense than water l Helium is less dense than air. l A ship is less dense than water

64 Density of water l 1 g of water is 1 mL of water. l density of water is 1 g/mL l at 4ºC l otherwise it is less