Ch. 1 Introduction, Measurement, Estimating

1.4 Measurement and Uncertainty; Sig Figs

1.4 cont There is uncertainty associated with every measurement When giving the result of a measurement, it is important to state the estimated uncertainty in the measurement i.e. The width of the board maybe written as 8.8 +/- 0.1 cm % uncertainty is the ratio of the uncertainty to the measured value x100% i.e. 0.1/8.8 x 100% = ~1%

Contextual Example 1-1 A friend asks to borrow your precious diamond for a day to show her family. You are a bit worried, so you carefully have your diamond weighed on a scale which reads 8.17g. The scale's accuracy is claimed to be +/- 0.05g. The next day you weigh the returned diamond again, getting 8.09g. Is this your diamond?

1.4 cont Significant Figures 23.32 cm = 4 Sig figs 80 km = 1 sig fig Count from the right to left, excluding any zeros as place holders 3.14150 = 5 sig figs 3.104156 = 7 sig figs 0.00045 = 2 sig figs 57630000 = 4 sig figs

1.4 cont The final result of multiplication and division should have only as many digits as the # with the least # of sig figs used in the calculation i.e. 11.2cm x 6.7cm = 75.04cm^2 which should be approx to 75cm^2 EX. The area of a rectangle 4.5cm by 3.25cm is? When adding or subtracting #s, the final result is no more accurate than the least accurate # used i.e. 3.6 – 0.57 is 3.0 (Not 3.03) To obtain the most accurate result, you should keep one or more extra sig figs throughout a calc, and round off only the final result

1.4 cont Note: calcs can give too few sig figs i.e. 2.5 x 3.2 = 8 on a calc when in fact it should read 8.0 EX. Do 0.00324 and 0.00056 have the same # of sig figs? EX. State the # of sig figs AND the # of decimal places A) 1.23 B) 0.123 C) 0.0123

1.4 cont Scientific Notation and % Error We commonly write #s in the power of 10 or scientific notation 36,900 = 3.69 x 10^4 0.0021 = 2.1 x 10^(-3) One advantage of sci not is that it allows the # of sig figs to be clearly expressed % error

1.5 Units, Standards, and the SI System The measurement of any quantity is made relative to a particular standard or unit, and this unit must be specified along with the numerical value of the quantity Standards must be readily reproducible Physical quantities can be divided into two categories: base quantities and derived quantities. A base quantity must be defined in terms of a standard All other quantities can be defined in terms of these 7 base quantities, called derived quantities i.e. Speed is derived from distance divided by time To define any quantity we can specify a rule or procedure, this is called operational definition

SI Base Quantities and Units Quantity Unit Abbreviation Length Meter m Time Second s Mass Kilogram kg Electric Current Ampere A Temperature Kelvin K Amt of substance Mole mol Luminous Intensity Candela cd

Metric SI Prefixes Prefix Abbreviation Value Yotta Y 10^24 Zetta Z 10^21 Exa E 10^18 Peta P 10^15 Tera T 10^12 Giga G 10^9 Mega M 10^6 Kilo k 10^3 Hecto h 10^2 Deka da 10^1 Deci d 10^(-1) Centi c 10^(-2) Milli m 10^(-3) Micro μ 10^(-6) Nano n 10^(-9) Pico p 10^(-12) Femto f 10^(-15) Atto a 10^(-18) Zepto z 10^(-21) Yocto y 10^(-24)

1.6 Converting Units 1 in = 2.54 cm OR 1 = 2.54 cm/in EX Convert 21.5 in to cm Units must cancel each other out leaving behind only the unit in question EX Convert 8000m to ft EX Convert 1.25 sq in to sq cm Hint: 1 in2 = 6.45 cm2

1.7 Order of Magnitude: Rapid Estimating A rough estimate is made by rounding off all #s to one sig fig and its power of 10, and after the calculation is made, again only one sig fig is kept. Such an estimate is called an order of magnitude estimate

1.8 Dimensions and Dimensional Analysis When we speak of the dimensions of a quantity, we are referring to the type of units or base quantities that make it up The formula for a quantity may be different in different cases, but the dimensions remain the same Area of a triangle 1/2 x b x h Area of a circle πr2 The formulas are different, but the dimensions of the area are the same L^2 When we specify the dimensions of a quantity, we usually do so in terms of base quantities, not derived quantities i.e. Force has the same units as mass (M) times acceleration (L/T2), has dimensions of (ML/T2)

1.8 cont Dimensions can be used as a help in working out relationships, and such a procedure is referred to as dimensional analysis A simple rule applies here: we add or subtract quantities only if they have the same dimensions