# Metric System Math Review -Dimensional Analysis

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1 Metric System Math Review -Dimensional Analysis In 1960 the International Bureau of Weights and Measures (Sevres, France) adopted the "International System of Units", also known as "SI" units. 7 base units are used. All other SI units are derived from combinations of these base units: SI Base Units Length meter m Mass kilogram kg Time second s Electric Current ampere A Temperature Kelvin K Amount of Substance mole mol Luminous Intensity candela cd General Punctuation Rules: a) use space instead of comma to separate numbers into groups of three ex: m m b) use decimals instead of fractions to express partial units c) space between the numeric value and unit symbol d) do not use periods after symbols e) all abbreviations are singular f) follow capitalization rules as given in problems Metric Prefixes are combined with a unit to indicate that the unit has been multiplied by a certain power of 10. Prefixes are to be used to keep numerical values in tables, results, and specifications between 0.1 and example: N = 144 kn Six basic prefixes are used commonly. They will be used to demonstrate the basic principles. Less common prefixes are given in the metric appendix. The first example: kilo (abbreviated with lowercase 'k' This prefix represents 3 decimal places, usually associated with examples: 1 km = 1000 meters (1000 m) 2.3 kj = Joules (2 300 J) Another example: centi (abbreviated with lowercase 'c' This prefix represents 2 decimal places, usually associated with 0.01 examples: 100 cm = 1 meter (1 m) 2.3 E 2 cm = 2.3 meters (2.3 m) Another commonly used metric prefix is milli ('m') examples: 1000 ml = 1 liter (1 L) 2.3 E 3 ml = 2.3 liters (2.3 L)

2 Less common but used occasionally are the following: mega (M) (1 E 6) micro (µ) (1 E - 6) nano (n) (1 E - 9) The remaining prefixes and their multiplication factors can be seen in the metric appendix. Dimensional Analysis Examples Dimensional analysis is also called the factor-label method of problem solving. It is a way of setting up a problem in a constant fashion that breaks the problem down into simple steps. Each step is a ratio that must equal 1, thus canceling out some preceding unit. example problem: 4.4 km =? m solution: 4.4 km x 1000 m = m 1 km (note that the 'km' cancel and you multiply 4.4 times 1000 to get your answer) problem: 4.4 km =? cm solution: 4.4 km x 1000 m x 100 cm 1 km 1 m = 4.4 E 5 cm solution using scientific notation: 4.4 km x 1 E 3 m x 1 E 2 cm = 4.4 E 5 cm 1 km 1m problem without metric prefixes: 4.4 weeks =? seconds solution: 4.4 wk x 7 day x 24 hour x 60 minute x 60 seconds = 2.7 E 7 s 1 wk 1 day 1 hour 1 minute some points of interest: 1 E 3 m = 1 km place the 1 with the large unit (then all other numbers in conversion are greater than 1) use base units whenever possible must know number of decimal places in each prefix 1 cm 3 = 1 ml for all substances 1 cm 3 = 1 ml = 1 g for pure water only Metric Problems: the following problems solutions are part of the homework. Try them before you look at the solutions (no peeking). 4 g to cg 12 L to ml ml to µl kl to L 8.5 E - 5 mg to cg 1.85 E - 6 Mm to nm Solutions: 4 g x 1 E 2 cg = 4 E 2 cg 1 g

3 12 L x 1 E 3 ml = 1.2 E 4 ml 1 L 12.5 ml x 1 L x 1 E 9 µl = 1.25 E 7 µl 1 E 3 ml 1 L kl x 1 E 3 L = 68 L 1 L 8.5 E - 5 mg x 1 g x 1 E 2 cg = 8.5 E - 6 cg 1 E 3 mg 1 g 1.86 E - 6 Mm x 1 m x 1 E 9 nm = 1.86 E - 3 nm 1 E 6 Mm 1 m Metric Appendix Name Symbol Multiplication Factor exa E 1 E 18 petra P 1 E 15 tera T 1 E 12 giga G 1E 9 mega M 1 E 6 kilo k 1E 3 hecto h 1 E 2 deka da 1 E 1 deci d 1 E - 1 centi c 1 E - 2 milli m 1 E - 3 micro µ 1 E - 6 nano n 1 E - 9 pico p 1 E - 12 femto f 1 E - 15 atto a 1 E - 18 Common Conversions 1 mile = 5280 ft 1 mile = 1760 yd 1 m = 100 cm 1 in = 2.54 cm 1 slug = 14.6 kg 1 km = 1000 m 1 hour = 3600 s 1 km = mile 1 L = 1.06 qt1 km = mile 1 m = 3.28 ft 1 lb = 454 g 1 year = day

4 Significant Figures * There is a difference between pure numbers (as used in math classes) and measured quantities (such as 35.6 ml of acid). * Measurement tools often limit our ability to measure quantities. * We must be aware of the accuracy limits of each piece of lab equipment that we use and record our data to the proper number of significant figures. * You are always allowed one estimated figure in measurements. * When measured quantities are given to you, it is assumed that the proper number of significant figures were recorded. (all measurements in scientific notation must be recorded to the proper number of significant figures). * When using conversions, `1' is considered to be correct to several significant figures. Rule: All non-zero digits are significant cm [4] g [5] Rules for Counting Significant Figures Rule: Zeros between other significant figures are significant cm [5] g [5] g [ ] kg [ ] Rule: Zeros to the right of a decimal point and to the right of a number are significant cm [5] m [5] ml [4] 1.30 kg [ ] cm [ ] Rule: A zero standing alone to the left of a decimal point is not significant g [3] 0.5 m [ ] Rule: Zeros to the right of the decimal and to the left of a number are not significant mg [3] cm [3] kg [ ] m [ ] Rule: Zeros to the right of the last number but left of the decimal point may or may not be significant. This information is known only to the person that made the measurement. Use scientific notation when in doubt. The use of a bar over the last significant zero is acceptable as well as using the decimal point to indicate that all digits to its left are significant g [4] 3400 kg [2] ml [ ] 4.50 E 2 g [ ] Rounding Rules When rounding examine the digit to the right of the digit that is to be last (this number is called the trigger): a) if the trigger is less than 5, drop it and all the digits to the right of it b) if the trigger is more than 5, increase by 1 the number to be rounded C) if the trigger is 5, round the number so that it will be even m [4] = m m [5] = m m [3] = 3.79 m m [3] = 24.8 m

5 Scientific Notation Integral Powers of = 1 E 0 = 1 (1 x 10 0 is acceptable) 10 1 = 1 E 1 = = 1 E 2 = = 1 E 6 = = 1 E -1 = = 1 E -2 = = 1 E -6 = Proper Notation: Place the leftmost significant figure to the left of the decimal and display the rest of the figures (significant only) to the right of the decimal following rounding rules. There can only be 1 integer to the left of the decimal. example: m = 9.3 E 7 m in = 2.8 E -5 in Decimal points and a bar over the last significant zero will be demonstrated in class. Multiplying measured quantities using scientific notation: Rule: multiply the numbers and add exponents: example: (9.0 E 3 m) (7.0 E 2 m) = 63 E 5 m 2 = 6.3 E 6 m 2 Dividing in scientific notation: Rule: Divide the numbers and subtract the exponents: example: (4.5 E - 6 m 2 ) / (1.5 E -2 m) = 3.0 E -4 m Multiplication and Division using Significant Figures Rule: In multiplication and division, the result may have no more significant figures than the factor with the fewest number of significant figures. example: 2.52 m x m = m 2 but must be recorded as 2.52 m 2 (3 sig figs) example: 7540 m x 1.3 m = m 2 but must be recorded as 9800 m 2 (only 2 sig figs) Rule: When adding and subtracting numbers that come from measurements, arrange the numbers in columnar form. The final answer can contain only as many decimal places as found in the measurement with the fewest number of decimal places. example: m m = m m m (2 decimal places)

6 Directions: Perform all the operations using your calculator and record the answer with the proper number of significant figures. See Mr. Jones for answers. 1. ( E 2) (5.61 E -4) (9.87 E 5) (3.1) (1.171 E 4) 2. ( E 2) ( E 5) (8.22 E 7) ( E -3) (9.711 E 4) 3. ( E 11) (5.6 E 4) (2.173 E 8) 4. (1.745 E -2) (9.51 E -7) (16.21) (9.346 E -10) 5. ( E 5) ( E 7) (9.5 E 1) Word Problems using Significant Figures Directions: Solve the following word problems and record the answer using the proper number of sig figs. Use for π. See Mr. Jones for answers. 1. The volume of a cylinder is: V = π r 2 L What is the volume of a cylinder that has a radius of m and a length of 2.0 m? 2. The 100 sheets of a textbook measure m. What is the thickness of only 1 of those sheets? 3. formula: t = d/v How long does it take light traveling at E 8 m/s to travel across a room that is 6.2 m long? 4. formula: a c = 4π r/t 2 Find the centripetal acceleration (a c ) when the radius is m and the period (T) is 0.64 s. 5. formula: d = 1/2 a t 2 What is the distance traveled by a body from rest when the gravitational acceleration (a) is m/s 2 and the fall time (t) is s? 6. I massed 150 tacks. The total mass was g. What is the mass of a single tack? Word Problems Using Significant Figures Rule: When calculating the number of significant figures allowed in an answer, the following categories are NOT considered: Numbers representing a numerical count: example: 25 students, each carrying kg of sugar 25 x kg = kg

7 Numbers representing mathematical formulas: example: volume of a sphere = 4/3 π r 3 for sphere with radius cm 4/3 π = cm 3 (using for π) Numbers in a conversion: 1 km = 1000 m the 1 is considered to have several significant figures Sample problems: (not found in other practice problem sheets) 1. What is the perimeter of a rectangle with a length of 2.1 m and a width of 0.65 m? P = 2l + 2w 2. The period of an oscillating spring is T = 2 π(m/k). If the spring constant (k) is N/m and the mass (m) is 1.2 kg, what is the period? 3. How long does it take light traveling at E 8 m/s to travel across a room that is 6.2 m long? 4. The 100 sheets of a textbook measure m. What is the thickness of only 1 of those sheets? 5. The period of revolution for Uranus is days. How many hours is that? 6. If the thickness of 1 sheet of paper is cm, what is the thickness of 6 of those sheets? 7. The perimeter of a polygon is the sum of its sides. What is the perimeter of a polygon with sides of m, 5.97 m, 3.21 m, 0.2 m, and m? Answers available from Mr. Jones

8 Dimensional Analysis Dimensional analysis is also known as the factor-label method of problem solving. It involves the use of conversions to systematically change units. The following examples may help: 1.0 ft = cm 1.0 ft x 12 in x 2.54 cm = cm = 30. cm 1 ft 1 in 63 s = ns (nanoseconds) 63 s x 1 E 9 ns = 63 E 9 ns = 6.3 E 10 ns 1 s 1.00 year = s 1.00 yr x day x 24 hr x 3600 s = 3.2 E 7 s 1 yr 1 day 1 hr Expansion - Contraction Coefficients Almost without exception, solids expand upon heating and contract on cooling. The table given below shows the change in length (expansion or contraction) when a unit length (1 m or 1 ft, etc) of a material is warmed or cooled one Celsius degree. This fractional change in length for a temperature rise of 1 is called the COEFFICIENT OF LINEAR EXPANSION. For example, 1.00 m of steel increases 2.00 E-5 m when it is warmed 1 C. Consider a bridge 1.00 E 3 m long. For every increase in temperature of 1 C every meter of the bridge increases by 2.00 E-5 m. This results in a total of 2.00 E-2 m increase in length. When the temperature increases by 10.0 degrees the increase in length will be 10 times as much. And between winter and summer, a change of temperature of as much as 40.0 may take place, increasing the length of this bridge by m (1.00 E 3 m x 40.0 C x 2.00 E-5 / C ). If the engineer did not allow for expansion, the force exerted by this expansion (or contraction in winter) would result in buckling of steel girders. In metal rods or wires, the expansion or contraction that engineers are concerned about is essentially in one dimension -- length (if you were dealing with a block of metal, you could observe expansion or contraction in three dimensions). In a liquid, however, the expansion or contraction in three dimensions in always important. Therefore the figures in the table for the expansion of liquids express the COEFFICIENT OF CUBICAL EXPANSION - the change in volume for a unit volume (liter, gallon, etc.) of a liquid changing one Celsius degree in temperature. You cannot easily observe the effects of temperature change on gases unless the gas is sealed in a container. The gases of the atmosphere are free to move when expanding or contracting. However, experimentation has shown that unlike solids and liquids, all gases expand and contract to the same extent with a change in temperature. That is, all gases have the same coefficient of expansion or contraction.

9 The behavior of substances as the temperature changes gives us another means of identifying them. Solids melt and liquids boil; if they are pure substances, they melt or boil at definite temperatures. But whether or not these changes occur, matter in all states expands or contracts with a change in temperature, and the extent of this change can be measured. Melting points, boiling points, and the amount of expansion or contraction are three properties useful in distinguishing among substances. but some substances neither melt nor boil. Still other properties are needed to distinguish between them. Problems: 1. Using the tables (below), determine what change in length or volume results a. an aluminum rod 1.00 m long is warmed 1.00 C: b. a glass rod 5.00 m long is warmed 2.00 C: c m of platinum wire are cooled 5.00 C: d L of grain alcohol are cooled 10 C: e L of benzene is warmed 5.00 C: 2. Using the tables, to what temperature must a steel rod 1.00 m long at 20.0 C be heated to increase in length by 1.00 E - 4 m? 3. An unmarked volume of a liquid was warmed from 20.0 C to 30.0 C. This resulted in an increase of L. The coefficient of cubical expansion of the liquid was 5.10 E-4/ C. What was the original volume of the liquid? Table of Coefficients of Linear Expansion - SOLIDS Table of Coefficients of Cubical Expansion - LIQUIDS aluminum 2.3 E - 5/ C ethanol 1.12 E - 3/ C platinum 9.0 E - 6/ C benzene 1.24 E - 3/ C glass 9.0 E - 6/ C mercury 1.8 E - 4/ C steel 2.0 E 5/ C water 2.1 E 5/ C Calculator Lab Can you use each of these functions using your calculator? 1) clear entry vs all clear 2) change sign 3) () 4) scientific notation 5) inverse (inv on some, 2 nd function on others) 6) square roots and squares 7) Y to X power 8) memory (store, recall, clear memory) 9) Pi 10) sin, cos, tan Sample problems (answers given to proper # of sig figs) 1) (6.29 E 5) (4.39 E - 7) = 2.76 E - 1 (0.276) 2) (7.80 E - 4) / (9.55 E 12) = 8.17 E - 17

10 3) (3.243 E 5) = (1.812 E 4) (2.530 E - 1) 4) 3 + (4 x 3) x 2 = 27 5) 3 + (4 x 3) x 2 3 = 99 6) 3 + (4 x 3) x = 106 7) (6.07 E - 2) 2 = 3.68 E - 3 ( ) 8) (5.45 E 2) 3 = 1.62 E 8 9) (6.38 E - 4) 7 = 4.30 E ) (6.291 E 3) 1/3 = ) (2.173 E 4) 1/9 = ) [1/3 x 6/15 + 4/13] 3 = ) [4 x (6 / 5)] 1/3 = 6.94 E - 2 (0.0694) (12 x 4) + (2 / 6) 2 14) sin 9.614º = ) tan 15.71º = ) sin = 10.05º 17) inv sin = 78.75º 18) 8.33 (4-5.2) / (( ) x 0.32) = ( ) - (1.71 x 2.01) Accuracy, Precision, Average Deviation, Error Mathematics deals with pure numbers, and pure numbers are precise, or exact. In science, getting precise numbers is difficult unless, for example, you divide a number by 2, which is exact, or multiply by a whole number. However, most data is not exact. It might be possible to get a precise value for the number of people in a room, bit if there are 50 people moving about it might be difficult. You might count them 3 or 4 times and get a different value each time. Your value might not be very accurate or precise. In science, there is often a wide range of acceptable values, and few numbers are exact. Accuracy is a measure of how close a number is to the actual value (how far from the actual value a particular measurement falls). In many experiments, accuracy cannot be determined; only an estimate of the accuracy can

11 be found, because the actual value is not known. Precision, on the other hand, refers to how close repeated measurements are to each other (a measure of the variation present in a set of readings). see drawing on board here Although standard deviation is a more accurate method of finding the error margin we will use the average deviation method because it is relatively easy to calculate. To Find Average Deviation (precision of measurement) 1. Find the average value of your measurements. 2. Find the difference between your first value and the average value. This is called the deviation. 3. Take the absolute value of this deviation. 4. Repeat steps 2 and 3 for your other values. 5. Find the average of the deviations. This is the average deviation. The average deviation is an estimate of how far off the actual values are from the average value, assuming that your measuring device is accurate. You can use this as the estimated error. Sometimes it is given as a number (numerical form) or as a percentage. To Find Percent Error 1. Divide the average deviation by the average value. 2. Multiply this value by Add the % symbol. Sample Problem: A man wants to see if his car gets the number of miles per gallon (mpg) claimed by the dealer. he took data for five fill ups and found that he got 27, 33, 28, and 32 mpg. The car manufacturer stated that he should get 32 mpg. Find if the stated mpg agrees with the advertised value. Solution: The average value is found to be 31 mpg. ( ) / 5 = 31 mpg. Precision is found to be plus or minus 3, because ( ) / 5 = 2.8, which is rounded to 3, as this is just an estimate of how far off the answer is. This give a value of 31, plus or minus 3 mpg or between 28 and 34 miles per gallon. It shows the car is doing OK. Average Deviation Problems 1. On five different tankfuls of gas a pickup got 12, 15, 16, 12, and 15 mpg. Find the average mpg of this truck. 2. Find a value that gives an estimate of the error in gas mileage of the truck in problem Find the percent error for mileage given the numerical error that you found in problem Suppose that instead of using a truck, the person in the problem had used a car that got 39, 45, 47, and 41 mpg on four tanks of gasoline. Find the average value for the mpg, and estimate how much range in value is expected. 5. Find the estimated percent error in problem 4.

12 6. You walk down a football field and find that in 100 yards, you have taken 150 steps. You then measure a distance by walking 240 steps. How far have you gone? Once the estimated error has been found for a measurement two rules can be used in the calculation of an error estimate when doing mathematical operations: 1. When adding or subtracting, add the numerical error. 2. When multiplying or dividing, add the percent error. Counting Significant Figures 1) 967 2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 9 25) 90 26) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 16

13 Expressing the Very Large and the Very Small Instructions: Express each of the following numbers using scientific notation or change from scientific notation to standard notation. 1) 325 2) 70 3) ) E 3 5) ) ) E - 3 8) ) ) 4 E 3 11) ) ) ) ) 1.7 E ) 3 17) ) E ) 215

14 20) ) ) ) ) ) ) ) ) ) E 2 30) ) ) ) ) ) ) e 4 37) ) ) )

15 Math Practice Quiz 1 1. Correct the following by using the proper punctuation: a. 630, 000, 000 m b. 43 1/3 foot c. 6875ml d secs e Kg f. 545 in./sec. 2. Complete the following: a. 1 km = m b. 1 m = km c. 10 mg = g d. 10 g = mg e. 1 m = µm f km = mm g mm = km 3. In the brackets indicate the number of significant figures in the given measurement: a m [ ] b A [ ] c s [ ] d K [ ] e mol [ ]

16 f kg [ ] g m [ ] h K [ ] 4. Round each of the following measurements to the number of significant figures indicated in the brackets: a m [ 4 ] b mm [ 3 ] c nm [ 4 ] d km [ 2 ] e m [ 4 ] 5. Perform the indicated operation and record the answer to the proper number of significant figures: a m x 3.1 m = b m x 240 m = c. 3.5 m 2 / 1.50 m = d m m = e m m = f cm 2 / 9 cm =

17 6. List the proper standard base unit in the SI metric system: a. time: b. temperature: c. length: d. mass: e. amount of substance: 7. Express the following as a whole number (or decimal number if less than 1): a E 3 m = b E 3 m = c E 2 m = 8. Express the following in proper scientific notation: a m = b m = c m = 9. Perform the indicated operation and record the final answer to the proper number of significant figures: a. (1.6 E 24 g ) (4.60 E 30 mol/g) = b. (6.5 E 2 m) (2 E 6 m) = c. (2.80 E 5 m) (6.2 E 5 m) =

18 d. (9.46 E 10 g ) / (5.0 E 4 cm 3 ) = e. (3.4 E 4 cm 2 ) / (2.00 E 4 cm) = Practice Problems: Counting sig figs (answers given): cm [2] kg [2] g [3] s [3] L [3] g [4] s [4] m [7] s [3] s [5] Scientific Notation (Exponential Notation) (1 E 9) (1 E-7) (8.5 E 7) (4.3 E15) (1.54E-4) E 4 (95 000) E-9 ( ) E 4 ( ) E 9 ( ) E-1 (0.11) Signficant figures with math cm cm (14.4 cm) g g (44.1 g) ml ml ml (7.5 ml) 4. 3 cm x 4 cm (10 cm 2 ) mm x 1.22 mm (7.9 mm 2 ) 6. 0 cm x 15 cm (400 cm 2 ) m x 283 m ( m 2 ) m 2 / m (0.22 m) mm 2 / mm (3.733 mm) cm 2 / 30 cm ( cm or 1 E 4 cm) Use of Scientific Formulas The magnifying power of a telescope can be determined using the formula below. In the formula, F, represents the focal length of the larger lens and f, the focal length of the smaller lens. What is the magnifying power of a telescope in which the larger lens has a focal length of 62.5 cm and the smaller lens, a focal length of 50.mm? M = F/f Answer = 12.5X

19 Using the formula given above, find the magnifying power of the large telescope at the Yerkes Observatory if the large lens in that telescope has a focal length 18.9 m and the small lens has a focal length of 75.0 mm. Answer = 252X Metric Conversions: 4 g to cg (400 cg) 3 m to km (0.003 km) 12 L to ml ( ml) 6 m to nm (6 E 9 nm) 9 cm to m (0.09 m) 15 ml to L (0.015 L) 2.3 km to m (2 300 m) 10 µm to m (1 E -5 m) 12.5 ml to µl (1.25 E 4 µl) 55.8 mm to nm (5.58 E 7 nm) 0.6 cg to mg (6 mg) kl to L (68 L) 8.5 e -5 mg to cg (8.5 E -6 cg) km to mm ( mm) 1.85 E -6 µm to nm (1.85 E -3 nm) Metric-English Conversions: 3.0 yd to m (2.7 m) 8.0 ft to m (2.4 m) 2.5 kg to lb (5.5 lb) 250 cm to in (98 in) 11 lb to kg (5.0 kg) 4.2 in to cm (11 cm) 22 g to oz (0.78 oz) 7 mile to km (10 km 11.3 km really) 8 lb to cg (4 E 5 cg cg really) 24 m to yd (27 yd) 162 km to mile (101 mile) 5.5 E -3 yd to mm (5.1 mm) 325 mm to ft (1.07 ft) mm to ft ( ft) mg to lb ( lb) L to qt ( qt) gal to ml (473 ml) m to in (90.75 in) qt to L (27.39 L) yd to m (0.721 m)

20

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