Introductory Concepts. Units, dimensions, and mathematics for problem solving

Transcription

1 Introductory Concepts Units, dimensions, and mathematics for problem solving

2 Unit Conversion What is the value of S in cm per second? S = 5x10 3 furlongs fortnight Conversion Factor: number with 2 different units of measure Solution requires convenient placement of conversion factors: S = 5x10 3 furlongs fortnight x 1 mile 8 furlong 5280 feet x mile 12inches x foot x 2.54cm inch 1 fortnight x 14days x 1day 24hours 1hour x 3600sec = 83.15cm sec

3 Scientific Notation 1. Format : N x 10 x where 1 N< 10 and x = the location of the decimal place in the original number 2. Indication of precision is by the number of values after the decimal place. 3. Mathematics is algebraic: a. X a * X b = X a+b b. X a / X b = X a-b c. (X a ) b = X a*b d. b X a = X a/b

4 Exponent Arithmetic Multiplication: X a * X b = X a+b 3.0 x 10 3 * 4.0 x 10 5 = (3.0*4.0) x = 12 x 10 8 = 1.2 x 10 9 Division: X a / X b = X a-b / x 10-4 / 1.6 x /1.6 = 5 ; 10-4 / 10-2 = 10-4-(-2) = 10-2 = 5 x 10-2

5 Dimensional Analysis Consider the equation: y = x + a( w + z) What do we know about the units on these quantities? (1) The units on x and y are the same. (2) The units on w and z are the same. (3) The units on aw and az are the same as on x and y. Thus, if y has units of length (cm), x has units of cm. If the units on w and z are grams, the units on a must be cm gram

6 Dimensional Analysis (Continued) Consider the equation for the energy change occurring when an object is heated : E = T T 2 2 T C [ T ] + A [ ]. If ΔE has units of joules (J), What are the units of C? What are the units on A? K x (J/K) = J K 2 x (J/K 2 ) = J What can we say about the following equation? E = 2 2 C[ T T + T T ]

7 Transcendental Functions Let Z=A+B (10) x y How are the units of Z, A, and B related? A and B units = Z units How are the units of x and y related? x units = y units Let: Z=A log[x/y] or Z=A e x/y How are the units on Z and A related? Z units = A units How are the units on x and y related? x units = y units If we write, Z = A+ log( x ) y What are the units on A and Z? Z units = A units

8 Statistical Analysis Let us assume that a quantity x is measured 6 times with the following results: Set A Set B x1= x2= x3= x4= x5= x6= N Define: Average Value of x = < x >= Σ x i / n < X < x > i=1 =( )/6 = for Set A > is also called the first moment of the distribution, or the expectation value. < x > For Set B, the result is =2.526

9 Spread of a Distribution Although < x > is the same for both Set A and Set B, it is clear that the two values are not equally precise. There is a lot more spread in the values for Set B. Hence, we say these values are less precise. Root Mean Square Deviation from the Mean A measure of the spread of a distribution is given by σ, where we define σ to be n i= 1 σ = ( x < x> ) i [ nn ( 1)] 2 For set A, we have: σ ={[( ) 2 + ( ) 2 +( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 ]/(6)(5)} 1/ σ a = = ± σ 30 b =

10 Significant Digits Multiplication and Division: The number of significant digits in the answer equals the number in the factor containing the fewest number of significant digits. x = ( 5.2)(11.621) = = 60 ± 1 that is, we expect x to be in range Addition and Subtraction: 1.All digits to the left of the decimal point are significant if all terms have only significant digits to the left of the decimal point. Example: =11, significant digits in the answer.

11 2. If any # lacks significant digits to the left of the decimal point, the answer lacks significance in positions where one or more terms in the sum do not have significant digits = 11,797. Example: but the last digit is not significant:i.e. answer is The number of significant digits to the right of the decimal point for the answer is equal to the number of significant digits to the right of the decimal point in data with the least significant places. Example: = But there are only 2 significant digits to the right. Hence, we report the answer as 10, [7 significant digits]

12 When subtraction occurs, significance can be easily lost: Example: =0.001 (1 significant digit only) and for: = , only the first digit to the right is significant: answer = 0.0 which contains no significant digits.

13 The Number of Significant Figures in a Measurement Depends Upon the Measuring Device

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15 Common SI-English conversion factors Quantity SI Unit SI Equivalent English Equivalent Length 1 kilometer(km) 1000(10 3 )m 0.62miles(mi) Volume 1 meter(m) 100(10 2 )m 1.094yards(yd) 1000(10 3 )mm 39.37inches(in) 1 centimeter(cm) 0.01(10-2 )m in 1 kilometer(km) 1000(10 3 )m 0.62mi 1 cubic meter(m 3 ) 1,000,000(10 6 ) cubic centimeters 1 cubic decimeter (dm 3 ) 1 cubic centimeter (cm 3 ) 1000cm cubic feet (ft 3 ) gallon (gal) quarts (qt) English to SI Equivalent 1 mi = 1.61km 1 yd = m 1 foot (ft) = m 1 in = 2.54cm (exactly!) 1 mi. = 5,280 ft. 1 ft 3 = m 3 1 gal = dm 3 1 qt = dm dm fluid ounce 1 qt = cm 3 1 fluid ounce = 29.6 cm 3 Mass 1 kilogram (kg) 1000 grams 2,205 pounds (lb) 1 gram (g) 1000 milligrams ounce(oz) 1 (lb) = kg 1 lb = g 1 ounce = g

16 The Freezing and Boiling Points of Water

17 Temperature Scales and Interconversions Kelvin ( K ) - The Absolute temperature scale begins at absolute zero and only has positive values. Celsius ( o C ) - The temperature scale used by science, formally called centigrade and most commonly used scale around the world, water freezes at 0 o C, and boils at 100 o C. Fahrenheit ( o F ) - Commonly used scale in the U.S. for our weather reports; water freezes at 32 o F,and boils at 212 o F. T (in K) = T (in o C) T (in o C) = T (in K) T (in o F) = 9/5 T (in o C) + 32 T (in o C) = [ T (in o F) - 32 ] 5/9

18 Converting Units of Temperature PROBLEM: A child has a body temperature of C. (a) If normal body temperature is F, does the child have a fever? (b) What is the child s temperature in kelvins? PLAN: We have to convert 0 C to 0 F to find out if the child has a fever and we use the 0 C to kelvin relationship to find the temperature in kelvins. SOLUTION: (a) Converting from 0 C to 0 F 9 5 (38.70 C) + 32 = F (b) Converting from 0 C to K C = 311.8K

19 SI - Base Units Physical Quantity Unit Name Abbreviation mass length kilogram meter kg m time second s temperature kelvin K electric current ampere A amount of substance mole mol luminous intensity candela cd

20 Common Decimal Prefixes Used with SI Units Prefix Prefix Symbol Number Word Exponential Notation tera T 1,000,000,000,000 trillion giga G 1,000,000,000 billion 10 9 mega M 1,000,000 million 10 6 kilo k 1,000 thousand 10 3 hecto h 100 hundred 10 2 deka da 10 ten one 10 0 deci d 0.1 tenth 10-1 centi c 0.01 hundredth 10-2 milli m thousandth 10-3 micro µ millionth 10-6 nano n billionth 10-9 pico p trillionth femto f quadrillionth 10-15

21 Determining the Number of Significant Figures PROBLEM: For each of the following quantities, underline the zeros that are significant figures(sf), and determine the number of significant figures in each quantity. For (d) to (f) express each in exponential notation first. (a) L (b) g (c) ml (d) m (e) 57,600. s (f) cm 3 PLAN: SOLUTION: Determine the number of sf by counting digits and paying attention to the placement of zeros. (a) L 2sf (b) g 4sf (c) ml 5sf (d) m (e) 57,600. s (f) cm 3 (d) 4.715x10-5 m 4sf (e) x10 4 s 5sf (f) 7.160x10-7 cm 3 4sf

22 Rules for Significant Figures in Answers 1. For addition and subtraction. The answer has the same number of decimal places as there are in the measurement with the fewest decimal places. Example: adding two volumes 83.5 ml ml ml = ml Example: subtracting two volumes ml ml ml = ml

23 Rules for Significant Figures in Answers 2. For multiplication and division. The number with the least certainty limits the certainty of the result. Therefore, the answer contains the same number of significant figures as there are in the measurement with the fewest significant figures. Multiply the following numbers: 9.2 cm x 6.8 cm x cm = cm 3 = 23 cm 3

24 Issues Concerning Significant Figures Electronic Calculators be sure to correlate with the problem FIX function on some calculators Choice of Measuring Device graduated cylinder < buret pipet Exact Numbers numbers with no uncertainty 60 min = 1 hr 1000 mg = 1 g These have as many significant digits as the calculation requires.

25 Rules for Rounding Off Numbers 1. If the digit removed is more than 5, the preceding number increases by rounds to 5.38 if three significant figures are retained and to 5.4 if two significant figures are retained. 2. If the digit removed is less than 5, the preceding number is unchanged rounds to if three significant figures are retained and to 0.24 if two significant figures are retained. 3.If the digit removed is 5, the preceding number increases by 1 if it is odd and remains unchanged if it is even rounds to 17.8, but rounds to If the 5 is followed only by zeros, rule 3 is followed; if the 5 is followed by nonzeros, rule 1 is followed: rounds to 17.6, but rounds to Be sure to carry two or more additional significant figures through a multistep calculation and round off only the final answer.

26 Significant Figures and Rounding PROBLEM: Perform the following calculations and round the answer to the correct number of significant figures. (a) cm cm cm (b) 1 g 4.80x10 4 mg 1000 mg cm 3 PLAN: In (a) we subtract before we divide; for (b) we are using an exact number. SOLUTION: (a) cm cm 2 = cm cm 2 = cm cm (b) 4.80x10 4 mg cm 3 1 g 1000 mg = 48.0 g cm 3 = 4.16 g/ cm 3

27 Precision and Accuracy Errors in Scientific Measurements Precision - Refers to reproducibility or how close the measurements are to each other. Accuracy - Refers to how close a measurement is to the real value. Systematic error - Values that are either all higher or all lower than the actual value. Random Error - In the absence of systematic error, some values that are higher and some that are lower than the actual value.

28 Error Analysis Precise and Accurate Precise and in-accurate Lower precision but average is accurate In-precise and inaccurate Systematic error Random error