Measuring and Calculating Chapter 3

Outline Measuring and Units Measurements Mathematics and Measurements Orderly Problem Solving

What is Measuring? Measurement is the act of comparing an unknown quantity to a standard unit A dimension is a frame of measurement Length, mass, volume, time, and electrical charge are a few examples A unit has to be defined in order for something to be measured A measurement consists of two parts: A number A unit in which the measurement was taken. (Only like units can be added/subtracted but any units can be multiplied or divided when needed)

Metric Systems of Measurement The metric system is a measuring system based on a decimal scale, which just means the base units are by power of tens The widely accepted metric scale is called the International System of Units or SI In the 18th Century a need for a more standardized scale spurred the birth of the metric scale

SI Units Dimension Dimensional symbol SI Unit Unit Symbol Length 𝑙 There are 7 base units in the SI which covers basic dimensions: Dimension Dimensional symbol SI Unit Unit Symbol Length 𝑙 Meter m Mass M Kilogram kg Time t Second s Temperature T Kelvin K Number of particles n Mole mol Electrical Current l Ampere A Light Intensity IL Candela cd Look at the tables on page 50 and 51

SI Unit Prefixes The SI allows scientists to size units so that they are convenient to use In order to change the size of the units, SI adds prefixes that represent powers of 10 to make the unit either larger or smaller KNOW THE TABLE ON PAGE 52

Conversion Factors Unit conversion consists of multiplying the measurement by a conversion factor A conversion factor is a fraction that contains both the original unit and its equivalent value in a new unit Example: Convert 2.350 liters to milliliters I L= 1000 mL

Bridge Notation Bridge Notation is a special notation for multiplying and dividing several measurements together at the same time. Example: How many seconds are in 1 year

More Examples Convert the following using bridge notation: 132 540 cm to kilometers 350 mg to kilograms

Accuracy There will always be uncertainty in measurements as long as humans are the ones taking measurements The accuracy of a measurement is a numerical evaluation of how close the measured value is to the actual or accepted value of the dimension measured Whenever you measure something, you usually take a measurement knowing what it is supposed to be. By knowing the “accepted value” you can see how far off you are by calculating the percent error Percent error = observed value −accepted value accepted value ∗100% Example on page 56

Precision Precision indicates how repeatable a measurement is or how exactly one can make a measurement Precision is counting something 5 different times and getting the same number every single time….however, it doesn’t mean that what you counted was right Accuracy and precision are not the same thing!

Significant Digits When taking a measurement, the last digit is always the uncertain or least significant digit because when we take a reading we are often the most unsure about where that measurement falls The significant digits in a measurement are only those that are known for certain plus one estimated digit When reading a thermometer you always say “wellllll it could be .1 or maybe .0

Significant Digits Rules Significant digits apply only to measured data SD do not apply to counted or pure numbers SD do not apply to ratios that are exactly 1 by definition All nonzero digits are significant 25.4 mL = 3 SDs 13.78 g = 4 SDs All zeros between nonzero digits are significant 1003 cm = 4 SDs 1.09 g = 3 SDs Pure numbers are those without units and counted numbers are those that are exact 5 balloons

Significant Digit Rules Decimal points define significant zeros If a decimal point is present, all zeros to the right of the last nonzero digit are significant 20.0 s = 3 SDs 250.00 = 5 SDs If a decimal point is not present, no trailing zeros are significant 2500 mL = 2 SDs In decimal numbers, all zeros to the left of the first nonzero digit are not significant. They are placeholders 0.075 kg = 2 SDs 0.0010 s = 2 SDs

Significant Digit Rules Significant zeros in the one’s place are followed by a decimal point 2500. mL = 4 SDs The decimal factor of scientific notation contains only significant digits 2.50 x 103 mL = 3 SDs

Significant Digit Examples For each of the following determine the number of SDs 9.370 kg 63 000 g 705.06 mL 12 cookies 0.0001 s 2300.000

Calculations with Measured Data Measured data must be the same kind of dimension and have the same units before that can be added or subtracted The sum or difference of measured data cannot have greater precision than the least precise quantity in the sum or difference Example: Add 50.23 m, 14.678 m, and 23.7 m

Calculations with Measured Data The product or quotient of measured data cannot have more SDs than the quantity with the fewest SDs Example: What is the area of a piece of paper that is 11.5 cm ling and 5.5 cm wide? The product or quotient of a measurement and a counted number, conversion factor, or defined value has the same number of decimal places, or same precision, as the original measurement 3 x 3.64 g = 10.90 g

Calculations with Significant Digits In compound calculations, do not round off at the intermediate steps

Steps in Problem Solving Read the problem Read through the problem to figure out what is being asked and what data you are given Determine the method of solution Look for key words to show you how to solve the problem Choose the specific tools to use Figure out what type of equations you will have to use Set up the problem, estimate, and calculate Write down the equation, plug in your values, cancel units, and start calculating Check and format Check to make sure you really answered the question. Once you’re sure, make sure there are the right amount of SDs and make sure the unit is present

Example in Problem Solving A lead fishing sinker has a mass of 51.0 g. If the density of lead is 11.34 g/cm3, what is the volume of the sinker?