CHAPTER 2 The Metric System ConversionsMeasurement Significant Digits Graphing

Measure The Room Lab Measure the length and width of the room in “shoe” units. Discuss differences. Why doe we use standards? Name US units volume, distance, …etc Name Metric “Base” units. Name Metric prefixes Place prefixes in order smallest to largest.

2.1 The Metric System and SI Why use the Metric System Based on powers of 10, convenient to use The Syste’me Interationale d’Unite’s = SI = The Metric System This is the standard system used throughout the world by scientists, engineers, and everyone else everywhere except US.

Why don’t we use the Metric System in the United States? Good question.

Base Units Length = meter (m) Mass = gram (g) → “standard” unit = kg Volume = liter (l) Time = second (s) Temperature = Kelvin (K) Amount of a substance = mole (mol) …etc

Definition of a kilogram The mass of a small platinum-iridium metal cylinder kept at a very controlled temperature and humidity.

Definition of a meter The distance traveled by light in a vacuum during a time interval of 1/ seconds.

Definition of a second The fraquency of one type of radiation emitted by a cesium-133 atom.

SI Prefixes Giga – G Mega –M kilo – k hecto – h deka – da1010 BASE UNITmeters/liters/grams/…etc deci – d centi – c milli – m micro – μ nano – n pico – p

Introduction to meter stick Meter (m) – dm – cm - mm

METRIC “STEP” SYSTEM & CONVERSIONS Convert 102m → mm 102m = mm Convert μm → m μm = m Metric Conversion Hand Out #1

METRIC “Step System” G M k da h BASE d m c n p G M k da h BASE d m c n p G M k da h BASE d m c n p For each step you go up, move the decimal point one place to the left. For each step you go down, move the decimal point one place to the right. μ

Conversions Cont. Length1 inch = 2.54 cm (exactly) Volume1 liter = qt Mass1 kg = 2.21 lbs Weight1 lb = 4.45N (Newtons) More useful conversions on back cover of text book

Dimensional Analysis Method = Factor Label Method Example #1 4km → in 4km x 1000m x 100cm x 1in = 1 1km 1m 2.54cm 1 1km 1m 2.54cm in

Example #2 26dam → yds 26dam x 1000cm x 1in x 1ft x 1yd = 1 1dam 2.54cm 12in 3ft 1 1dam 2.54cm 12in 3ft284.3yds

More Example Problems 3) 37hl → gal 37hl = gal 37hl = gal 4) mg → tons mg = tons mg = tons

5) oz → Gg oz = 13,269.16Gg oz = 13,269.16Gg 6) mg → tons mg = 1.03tons mg = 1.03tonsHint: Insert the units first to ensure that the units will cancel out leaving only the unit that you want to end up with.

Handouts #3 & #4 When using the dimensional analysis method it is very helpful to insert units first, then the proper numbers.

SCIENTIFIC NOTATION (and Calculators) Convert 276Gl → pl 276Gl = pl Convert 146ng → Mg 146ng = Mg

Q: Is it convenient to use these types of numbers? A: NO!!!!!!  Scientific Notation is used to represent these very large/small numbers.

Rules for Scientific Notation The numerical part of the quantity is written as a number between 1 and 10 multiplied by a whole-number power of 10. M = 10 n where: 1 ≤ M < 10 n is an integer n is an integer

If the decimal point must be moved to the right to achieve 1 ≤ M < 10, then n is negative (-). If the decimal point must be moved to the left to achieve 1 ≤ M < 10, then n is positive (+) = 1

Therefore written in proper scientific notation: pl = 2.76 x pl ng = 1.46 x Mg

Calculator Buttons In class examples of E, EE, and positive/negative exponents.

Addition & Subtraction If the numbers have the same exponent, n, add or subtract the values of M and keep the same n. 3.7 x x 10 4 = ( ) x 10 4 = 9.9 x 10 4

Example x x 10 7 = (9.3 – 4.1) x 10 7 = 5.2 x 10 7

If the exponents are not the same, move the decimal point to the left or right until the exponents are the same. Then add or subtract M. Example x x 10 5 = 2.1 x x 10 8 = ( ) x 10 8 = x 10 8 or

Example – x x 10 5 = 2100 x x 10 5 = ( ) x 10 5 = x 10 5 = x 10 8 Exactly the same as previous example

If the magnitude of one number is very small compared to the other number, its effect on the larger number is insignificant. The smaller number can be treated as zero. (9.99 x 10 3 = 9999) 7.98 x x 10 3 = x x 10 3 = ( ) x 10 3 = x 10 3 = 7.98 x 10 12

Multiplication Multiply the values of M and add the exponents, n. Multiply the units x 10 7 m x 6.17 x s = (4.37 x 6.17) x 10 (7 + 13) (m x s) = x ms = x ms

Division Divide the values of M and subtract the exponents of the divisor from the exponent of the dividend. Divide the units. 7.9 x 10 9 m x 10 6 m = 3.1 x 10 (9 -6) m (4-3) = x 10 3 m

Challenging Addition 8.9 x 10 5 m km = 8.9 x 10 5 m x 10 3 m = 8.9 x 10 5 m + 76 x 10 5 m = ( ) x 10 5 m = 84.9 x 10 5 m = 8.49 x 10 6 m or

Challenging Addition Cont. 8.9 x 10 5 m x 10 3 km = x 10 5 km x 10 3 km = 0.89 x 10 3 km x 10 3 km = 8.49 x 10 3 km 8.49 x 10 3 km = 8.49 x 10 6 m

Challenging Multiplication 2.7 x μl X 4.3 x cl = x cl X 4.3 x cl = ( x 4.3) x 10 (10-4) (cl x cl) = x 10 6 cl 2 = x 10 3 cl 2

Challenging Division 6.2 x 10 8 kg 4.2 x Mg 6.2 x 10 8 kg 6.2 x 10 8 kg = 4200 x kg = 4200 x 10 (8- -5) = x = 1.47 x 10 10

SECTION 2.2 Measurement Uncertainties

Comparing Results Three students measure the width of a sheet of paper multiple times. #1 18.5cm→19.1cm, avg=18.8cm ∴(18.8 ± 0.3)cm #2 18.8cm→19.2cm, avg=19.0cm ∴(19.0 ± 0.2)cm #3 18.2cm→18.4cm, avg=18.3cm ∴(18.3 ± 0.1)cm Q: Are the three measurements in agreement? A: Students #1 & #2 have measurements that overlap, both have measurements between 18.8cm→19.1cm ∴ #1 and #2 are in agreement. However, student #3 does not have any overlap with #1 or #2, ∴ there is no agreement between student #3 and/or #1 & #2.

Accuracy and Precission Precision = The degree of exactness with which a quantity is measured using a given instrument. Q: Which student had the most precise measurement? A: #3 18.2cm–18.4cm, all measurements are within ± 0.1cm.

Generally when measuring quantities, the device that has the finest divisions on its scale yields the most precise measurement. The precision of a measurement is ½ the smallest division of the instrument. Q: How precise is a meter stick? A: The smallest division on a meter stick is a millimeter(mm)  you can measure an object to within 0.5 mm.

Accuracy = How well the results of an experiment or measurement agree with an accepted standard value. If the accepted/standard value of the sheet of paper was 19.0cm wide, which student was the most accurate, least accurate? Most accurate = #2. Least accurate = #3.

When checking the accuracy of a measuring device use the Two-Point calibration method. #1 Make sure the instrument reads 0 when it should. #2 Make sure the instrument yields the correct measurement on some accepted standard.

Techniques of Good Measurements Measurements must be made carefully. Common source of error = reading an instrument when looking at it from an angle  read the instrument from directly above. Parallax = the apparent shift in position of an object when viewed from different angles.

Significant digits Significant Digits = the valid digits in a measurement. The last (estimated) digit is called the uncertain digit. All non zero digits in a measurement are significant.

A = 1.24m B = 0.23cm How many significant digits for A & B? A = 3 B = 2 Which is a more precise measurement? A is to the nearest cm B is to the nearest 1/100cm  B is the more precise measurement

ZEROS Q: Are all zeros significant? A: No Q: Which zeros are significant? m # of significant digits =? A: 3 significant digits, first 2 zeros only show the decimal place, the last one is significant, it indicates the degree of precision of the measuring device.

m Q: How many significant digits? A: ???????????? Cannot tell, it is ambiguous, you do not know what instrument was used to achieve this measurement, possibly 3, 4, 5 or 6 significant digits.

To avoid confusion rewrite # 186 km = 3 sig dig km= 6 sig dig km= 4 sig dig 1.86 x 10 5 m= 3 sig dig x 10 5 m= 6 sig dig Mm= 3 sig dig Gm= 3 sig dig

Rules to Determine # of Sig Dig 1. Nonzeros are always significant. 2. All final zeros after the decimal point are significant. 3. Zeros between two other sig dig are always significant. 4. Zeros used solely as placeholders are not significant.

EXAMPLES # Sig Dig = = = = = 2

Addition & Subtraction Perform the operation, then round off the least precise value involved = = is the least precise value  round off to ,one digit past the decimal point.

Multiplication & Division Perform the calculation, round the product or quotient to the factor with the least significant digit cm x 7.2cm = cm 2  33cm x 1.9 =  7.0

More Examples 29.4m ÷ 2.431s = m/s  12.1m/s = 

Examples Cont =  x x 0.62 x 145 =  (ambiguous) More correct 4.0 x 10 4

Examples Cont ÷ 0.37 =  More correct 1.7 x 10 5

Rounding with “5” Rounding off when “5” is the last digit & = = When an even number precedes the 5, round “down”, when odd number precedes the 5 round “up”

Section 2.3 Visualizing Data Graphing Data Independent Variable = The variable that is changed or manipulated directly by the experimenter. Dependent Variable = A result of a Δ the independent variable, AKA, the responding variable. The value of the dependent variable “DEPENDS” on the Δ the independent variable.

Rules for Plotting Line Graphs 1. Identify the IV and DV. The IV is plotted on the horizontal, x-axis, and the DV is plotted on the vertical, y-axis. 2. Determine the range of the IV to be plotted. 3. Decide where the graph begins, the origin (0,0) is NOT ALWAYS the starting point.

4. Spread the data out as much as possible. Let each division on the graph paper stand for a convenient unit. 5. Number and LABEL the horizontal axis. 6. Repeat steps 2-5 for the DV. 7. Plot the data points on the graph.

8. Draw the “BEST FIT LINE”, straight or a smooth curve, that passes through as many data points as possible. Keep in mind the best fit line may not pass through any points. Do not draw a series of straight lines that simply “connect the dots”. 9. Give the graph a title that CLEARLY tells what the graph represents.

Linear Relationships A graph where a straight line can be drawn through ALL the points. The two variables are directly proportional; as x increases, y also increases by the same % and as x decreases, y decreases by the same %.

Example of Linear Graph

Slope = Linear Graph y = mx + b m = slope, the ratio of vertical Δ to horizontal Δ b = y-intercept, the point at which the line crosses the y-axis and it is the y value when x = 0.

rise Δ y y f - y i rise Δ y y f - y i m = run = Δx = x f – x i Slope can be negative, y gets smaller as x gets bigger.

Example TRANSPARENCY EXAMPLES Calculate slope Which slope is greater

Nonlinear Relationships A graph that produces a smooth curved line. Sometimes a parabola where the two variables are related by a quadratic relationship: y = ax 2 + bx + c Also expressed as ax 2 + bx + c = 0

One variable depends on the square of the other. Sometimes produces a graph that is inversely proportional, the graph is a hyperbola. Inverse relationship y = a/x or xy = a As one variable increases the other variable decreases.