Unit Conversion and SI Units

Transcription

1 Trigonometry Unit Conversion and SI Units Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. LAST REVISED June, 2007

2

3 Statement of Prerequisite Skills Complete all previous TLM modules before completing this module. Required Supporting Materials Access to the World Wide Web. Internet Explorer 5.5 or greater. Macromedia Flash Player. Rationale Why is it important for you to learn this material? Each measurement device used in the technologies uses some form of units to report the measurements taken. The units that the device reports back are not always the units that are needed. If this is the case some kind of unit conversion is necessary. This module will introduce the student to the most common units used in Canada and many of the conversion factors that are likely to be used in the technologies. Learning Outcome When you complete this module you will be able to Manipulate measurement units including prefixes and conversions. Learning Objectives 1. Identify the SI basic units, commonly derived units and permitted units and state their uses. 2. State the commonly used prefixes for SI units and understand their meaning. 3. Convert units within the metric system. 4. Convert units from the Imperial System to the metric system and from the metric system to the Imperial System. Connection Activity Consider the different measurement devices you are familiar with. Some examples may include a tape measure, a weigh scale, a protractor, calipers, speedometer, decibel meter, ohmmeter and so on. Can you think of others? How tall are you? Did you answer in feet and inches? If so, what would you do if you were filling out a form that required your height in centimetres? Convert of course. You will find this conversion factor and many others commonly used in the technologies in this module. 1

4 OBJECTIVE ONE When you complete this objective you will be able to Identify the SI basic units, commonly derived units and permitted units and state their uses. Exploration Activity Measurement is a comparison, of a quantity with a standard unit. In the past many standard units existed which were neither uniform nor acceptable to all. In 1960 the modern metric system was adopted: it is identified universally by SI for (System International d'unites). It supercedes any previous metric system in existence. Canada has been phasing in SI since the early 1970's and by now most of our units of measure are metric. However, our old Imperial system will persist for a while because Canada's main trading partner is slow in changing to SI. Table 1 1. There are seven base units in S I. BASE UNIT SI ABBREVIATION USED FOR MEASURING metre m length *kilogram kg mass second s time ampere A electric current **Kelvin K temperature candela cd light intensity mole mol molecular substance *The kilogram is the only base unit that contains a prefix. The gram proved to be too small for practical purposes. **The base unit for temperature is the Kelvin, but in the everyday world the degree Celsius is used instead. All other SI units can be defined in terms of the seven base units; they are called derived units. Derived Units with Compound Names A derived unit with a compound name is one that contains two or more units in its name. Examples are "metre per second" and "kilogram per cubic metre". There are many units with compound names, some of them are listed in the table below. 2 QUANTITY UNIT(COMPOUND NAME) SYMBOL Area square metre m 2 Volume cubic metre m 3 Speed metre per second m/s Acceleration metre per second squared m/s 2 Density Kilogram per cubic metre kg/m 3 Momentum Kilogram metre per second kg m/s Luminance candela per square metre cd /m 2

5 Derived Units with Special Names Some derived units are given special names for the sake of simplicity. For example, the derived unit of power, expressed in terms of base units is the "kilogram metre squared per second cubed". This is simply called the "watt", familiar to us through the use of electric light bulbs and appliances. These special names honor some great scientists of the past. Table 1-3. Some Derived Units with Special Names QUANTITY UNIT SYMBOL DERIVATION EXPLANATION Force Newton N kg m/s 2 mass times acceleration Frequency Hertz Hz 1 /S reciprocal time Pressure Pascal Pa N/m 2 force per area Energy, Work Joule J N m force times displacement Power Watt W J/s energy per time Electric Charged Coulomb C A s current times force Electric Potential Volt V J/C energy per charge Electric Resistance Ohm Q V/A potential difference per current NOTE: The derivation of a unit is not necessarily unique. For example, the volt may be defined as 1 watt per ampere (W/A) Permitted Units Some units of measurement outside SI have been used on a world-wide scale and will continue to be accepted for use along SI. Table 1 4. Some permitted units and their abbreviations. UNIT ABBREVIATION USED FOR MULTIPLE OF BASE MEASURING day d time s minute min time 60 s hour h time 3600 s year a time s degree º plane angles rad (approx) (angle) litre l volume 0.001m 3 tonne t mass 1000 kg degree (temp) ºC temperature K hectare ha area m 2 3

6 Rules for Combining Units There are certain rules of style used with numbers and their units when used in the SI. The following rules listed are the basic ones that we should observe. Many others can be found in more detailed reference sources. 1. Symbols remain unaltered if the unit is in the plural. For example, either metre or metres is designated by the symbol m. 2. When designating square or cubic units, we use exponents in the designation. For example, we use m 2 rather than sq.m. 3. The multiplication of unit symbols is designated by a dot of multiplication. For example, m A means metre times Amp (ma would mean milliamp). 4. The divisions of units is indicated by a solidus ( / ) or a negative exponent. For example a metre per second is written as m/s or m s 1. The solidus (/) is more commonly used. 5. In long numbers, the digits are separated in groups of three by a space, starting at the decimal point. A space is not necessary for a four digit number. For example, we write (rather than or 28,000). Also, we write rather than ). We may write either 5324 or 5 234, but we should be consistent. (NOTE: when entering numbers into a computer, no spaces should be used). 6. Where brackets occur in a symbol, this indicates that the bracketed onion is to be computed first. For example, P the unit for thermal conductivity, the watt per metre Kelvin, is expressed as W/(m K). 7. The spelling of the unit symbols should be exactly as that specified in the SI. For example k is the prefix kilo whereas K is the symbol for the Kelvin temperature. 8. A period marker should not be used at the end of a unit unless the unit is at the end of a sentence. 4

7 Experiential Activity One 1. Which statement defines measurement best: a) Measurement is the process of finding the distance between objects. b) Measurement is a comparison of a quantity with a standard quantity. c) Measurement is used to describe the sizes of clothing. 2. Answer True or False. a) There have been metric systems around before (T/F) b) SI was adopted to standardize previous metric systems. (T/F) 3. Write the correct base unit beside the following quantities: QUANTITY BASE UNIT Time Temperature Light intensity Mass Length Electric current Molecular substance 4. Circle which are not base units of the following units: s k h mm kg ma cd m m/s N 5. What do the following compound units measure? Write the answer beside the unit. m 2 m/s 2 kg/m 3 m/s m 3 6. Why are derived units given special names? a) For the sake of simplicity b) To honor great scientists c) So they are easier to tell apart d) Names are easier to remember 7. What is the symbol for the units with special names, used in measuring the following quantities. Write the symbol in the space provided. Pressure Force Work Electric resistance Frequency Power 8. What do the following permitted units measure? Write the answer in space provided l h t ha ºC º (degrees) 9. Answer True or False. We may use capital or lower case letters to write the same SI unit symbols. ANSWER: (T/F) 5

8 Experiential Activity One Answers Exercise b 2. a) T b) T 3. second Kelvin candela kilogram metre Ampere mole 4. k, h, mm, ma, m/s, N 5. Area, acceleration, density, speed, volume. 6. a 7. Pa, N, J, Ω, Hz, W 8. Volume, time. mass, area, temperature, angle 9. F 6

9 OBJECTIVE TWO When you complete this objective you will be able to State the commonly used prefixes for SI units and understand their meaning. Exploration Activity The base units and derived units are often not convenient in size for many types of measurements. Since the metric system is a decimal or base 10 system, multiples or submultiples of 10 for the SI units are used. These multiples or submultiples are called prefixes. Table 2-1. Prefixes for SI Units NAME OF SYMBOL MEANING MULTIPLIER POWER OF 10 PREFIX tera T one trillion giga G one billion mega M one million kilo k one thousand hecto h one hundred deca da ten ones deci d one tenth centi c one hundredth milli m one thousandth micro μ one millionth nano n one billionth pico p one trillionth Examples of the use of prefixes: one megavolt = one million volts, that is...1 MV = V one kilowatt = one thousand watts...1 kw = W one centimeter = one hundredth of a metre...1 cm = 0.01 m one milligram = one thousandth of a gram...1 mg = g one microsecond = one millionth of a second...1 μs = s NOTE: a) The most commonly used prefixes are kilo, centi and milli. b) Mega and micro are often used in science and technology. c) Hecto, deca and deci are well accepted prefixes, but are used only in limited applications. To ensure that prefixes retain their identity when pronounced, the first syllable of each prefix is accented. In particular, kilometre should be pronounced with the accent on the first syllable, as is our custom with such units as kilogram, kilolitre and kilowatt. The accent on the second syllable generally applies to measuring devices, such as thermometer, speedometer, barometer and micrometer. 7

10 Rules Governing Prefixes When using prefixes, certain rules must be obeyed. Here are some of them: Symbols for prefixes are lower case letters, except those that are greater than "kilo" Only one prefix is allowed at a time. For example, m should be expressed as 1 Mm and not 1 kkm In general, in the expression of any quantity, a prefix should be chosen so that the numerical value lies between 0.1 and However, when similar quantities are compared, it is better to use the same prefix for all items even though some values may fall outside the 0.1 to 1000 range. 8

11 Experiential Activity Two 1. Select the choice that best answers the following question. Prefixes are used because: a) base units are often not convenient in size b) it is easier to work with prefixes c) the metric system is a base 10 system d) they have always been used e) they are a necessity to use with SI 2. Write the numeric multipliers for the prefixes in the space provided. (The first one is an example) d 0.1 m M k c μ G h 3. Write the exponent of the power of 10 in the space provided. (The first one is an example k = 10 3 M = 10 d = 10 μ = 10 c = 10 h = 10 G = 10 da = State the three most commonly used prefixes in descending order of size. 5. State the number of times the prefix k is bigger than the prefix given. Write the answer in the space provided. a) m b) c c) μ d) h 6. State the number of times the prefix m is smaller than the prefix given. Write the answer in the space provided. a) k b) c c) M d) h 7. Fill in the number in the space provided. (The first one is an example) a) 1 kg = 1000 g b) 1 cm = m c) 1 MPa = Pa d) 1 μs = s e) 1 hl = l f) 1 GJ = J 8. What is the largest prefix that uses a lowercase letter for its symbol? ANSWER: 9

12 Experiential Activity Two Answers 1. a 2. d m M k 1000 c 0.01 μ G H , 6, 1, 6, 2, 2, 9, 1 4. k, c, m b) 0.01 c) d) e) 100 f) k 10

13 OBJECTIVE THREE When you complete this objective you will be able to Covert units within the metric system. Exploration Activity Changing the units of measurement to a different set of units is called unit conversion. When changing units within the same system the term unit reduction is sometimes used. To change from one unit to another unit, a conversion factor is often used. A conversion factor is simply a 1 written in a special way. For example 5 = 1, 5 12 m = 1, 12 m Also, since 100 cm =1 m, we have 5.6 kg 5.6 kg = 1, 100cm = 1 and 1m etc. 1m 100cm = 1 Hence, we have two conversion factors for each set of data. The correct choice depends on which units are to be cancelled. EXAMPLE 1 Change 624 cm to metres Solution 624 cm = 624 cm make an equation 1m 1m multiply by 1 = so that units ( ) 624 cm = 624 cm 100 cm 100cm cancel calculate answer (to divide by cm = 6.24 m simply shift the decimal point 2 places to the left ) EXAMPLE 2 Change kg to mg Solution kg = kg make an equation 1000 g 1000g multiply by 1 = so that ( ) kg = kg 1kg 1kg kg cancel kg = 1000mg multiply by 1 = so that 1000g 1000 mg ( ) 1g g g cancel calculate answer (to multiply by simply shift the kg = mg decimal point 6 places to the right ) 11

14 EXAMPLE 3 Convert m to mm 1000 mm Solution m = ( m) = 12205mm 1m Make an equation and multiply by conversion factor to get required units in answer. A quicker way is to combine the steps shown above into a more compact form. EXAMPLE 4 Solution EXAMPLE 5 Convert km to cm 1000m 100cm km = ( km) = 21700cm 1km 1m Make an equation and multiply by 2 conversion factors to get required units in answer. Convert 3140 mm 3 to cm 3 1cm 1cm 1cm 3 Solution 3140 mm 3 = 3140mm 3 = 3.140cm 10mm 10mm 10mm Make an equation and multiply by the same conversion factor 3 times. This will cancel the mm 3 and answer will be in cm 3. EXAMPLE 6 Solution EXAMPLE 7 Convert 1250 N/cm 2 to kn/m N 1kN 100cm 100cm N/cm = = kn/m 2 cm 1000 N 1m 1m Make an equation and multiply by the same conversion factors to cancel units. It makes no difference which units are converted first. Convert kn/m 2 to kpa 1Pa 2 2 Solution 12500kN/m = ( 12500kN/m ) = 12500kPa 2 1N/m 1N From table 1-3, we see that 1 Pa =. Hence, the conversion factor 2 m 1Pa is. 2 1N m Here the prefix k did not need to be converted. Conversions within the metric system involve only moving the decimal point a number of laces. With a little practice, one can convert units in the metric system mentally using the following reasoning show in the next example. 12 Note: Use this method only if units to be converted are in the numerator only.

15 EXAMPLE 8 Solution EXAMPLE 9 Convert cm to km We want to change from cm to km. From: c = centi = 10 2, the exponent is 2 To: k = kilo = 10 3, the exponent is 3 Difference of exponents = 2 3 = 5 Because the difference is negative, the decimal is moved to the left 5 places cm = km Convert l to ml Solution There is no prefix for l but 1 l can be written as 10 0 l, since 10 0 = 1 EXAMPLE 10 Solution From: 1 = 10 0, exponent is 0 To: m = milli = 10 3, the exponent is 3 Difference of exponents = 0 ( 3) = 3 Because the difference is positive, the decimal is moved to the right 3 places l = 325 ml Convert 1250 mm 2 to cm 2 From: prefix mm 2 = (milli) 2 = (10 3 ) 2 = 10 6, the exponent is 6 To: prefix c 2 = (centi) 2 = (10 2 ) 2 =10 4, the exponent is 4 Difference of exponents = 6 ( 4) = 2 Decimal point is moved to the left 2 places mm 2 = 12.5 cm 2 NOTE: When there is more than one unit to be converted in a measurement, it is safer to use conversion factors as shown in examples 1 through 8. Converting units mentally should be left only to the simple cases. If only one unit is to be converted, and that unit is in the numerator, there is a way to check if answer is reasonable. This is done by observing the size of the units and the size of the numbers. One should always have on each side of the equal sign, either a bigger number and a smaller unit or a smaller number and a bigger unit. 13

16 For example, converting km to cm, we get km = cm Observe: smaller number bigger unit = bigger number smaller unit or S B = B S So, here we have on each side of the equal sign) something small and something big. If we had converted km to cm as follows, we would immediately see that the answer is wrong km = cm??? B B = S S We cannot have two bigs or two smalls on one side of the equal sign. EXAMPLE m has been converted to km by moving the decimal point the correct number of places and the following answer was obtained: 23.5 m = km Is the answer correct? Solution The answer is correct because on one side of the equal sign, there is a bigger number (23.5) and a smaller unit (m), and on the other side, there is a smaller number (0.0235) and a bigger unit (Ian) so we have: EXAMPLE 12 B S = S B and the answer is correct. 484 kg has been converted to g by moving the decimal point the correct number of places. The answer obtained was: 484 kg = g Is the answer correct? Solution The answer is wrong because on one side of the equal sign, there is a bigger number and a bigger unit, and on the other side, there is a smaller number and a smaller unit so we have: B B = S S and the answer is wrong. 14

17 EXAMPLE 13 The following units have been converted. Indicate whether or not the answer is correct: CONVERSON CORRECT OR REASON INCORRECT 23.5 MN = kn correct S B = B S mg = 173.6g wrong S S = B B 27.5 dm = 275 m wrong S S = B B 848 ma = A correct B S = S B m = 0.43 cm correct S B = B S NOTE: The above method of checking works well if only one unit is to be converted. If a measurement contains 2 or more units, as for example 25 kn/cm 2, this method of checking should not be used. 15

18 Experiential Activity Three 1. Convert the following measurements. a) 2.75 km = cm e) l = ml b) 256 s = ms f) A = ma c) 3650 Pa = kpa g) 275 mm = cm d) 3450 kn = MN h) MW = W 2. Convert the following units. a) 2.75 m 2 = cm 2 e) 625 g/m 3 = kg/m 3 b) 217 N/cm 2 = kn/m 2 f) 3560 mm 2 = m 2 c) 75.2 g/m 3 = kg/m 3 g) cd/cm 2 = kcd/m 2 d) N/m 2 = kpa h) Mg/m 3 = g/cm 3 3. The following units have been converted. Fill in the missing prefixes. a) 2.38 m = 238 m e) 37.5 μf = F b) 63.5 dm = 6350 m f) 275 hl = 27.5 l c) MPa = 703 Pa g) 2.86 MΩ = 2860 Ω d) 2.75 g = g h) 892 mm 2 = 8.92 m 2 4. Fill in the missing number or missing unit. a) 73.5 cm = m e) 84.3 = km b) A = 375 f) ka = 17.5 c) 136 = Mg g) 175 μs = ms d) kpa = Pa h) mm 3 = 3.72 cm 3 5. State the number of places the decimal place has to be moved when converting units in the metric system. Indicate the direction by a positive number if direction is to the right and by a negative number if the direction is to the left. a) Change from km to m, answer = 3 (This is an example) b) Change from ma to ka, answer = c) Change from MΩ to kω, answer = d) Change from μv to kv, answer = e) Change from mf to nf, answer = f) Change from kpa to GPa, answer = 6. Convert, using the appropriate conversion factors, the following: a) 76.3 m/s to km/hr b) 848 cm/min to m/s c) 9.80 m/s 2 to km/min 2 Show Me. d) 1000 kg/m 3 to g/l e) 5.52 g/m 3 to kg/m 3 16

19 7. Convert the following: (NOTE: Number of degrees Celsius = number of degrees Kelvin ) a) Change K to ºC b) Change 127 ºC to K c) 725 K/s to C/min d) ºC/h to K/min 17

20 Experiential Activity Three Answers 1. a) b) c) 3.65 d) 3.45 e) 12.5 f) g) 27.5 h) a) b) 2170 c) d) a) c b) m c) k d) k 4. a) b) ma c) kg d) 37.5 e) f) g) h) e) n f) k g) k h) c e) m f) MA g) h) a) 6 b) 3 c) 9 d) 6 e) 6 6. a) 275 b) c) 35.3 d) 1000 e) a) 152 b) c) d)

21 OBJECTIVE FOUR When you complete this objective you will be able to Convert units from the Imperial System to the metric system and from the metric system to the Imperial System. Exploration Activity Although Canada has adopted the SI units of measurement, the units previously used will remain with us for some time. Hence, it is necessary that we are able to change units from one system to another. Such unit change is called unit conversion. In order to be able to convert units, we must know or have at hand the necessary conversion factors. Some of the most common conversion factors are given below. Table 4-1 QUANTITY CONVERSION FACTORS Length 1 inch = 1 in = 25.4 mm 1 foot = 1 ft = 12 in = mm 1 yard = 1 yd = 3 ft = m 1 mile = 1 mi = 1760 yd = km Area 1 acre = ha 1 mi 2 = 640 acres Volume 1 fluid ounce = cm 3 1 gallon = 1 gal = dm 3 (or l) Mass * 1 pound =1 lb = kg 1 slug = kg Temperature C = 5/9(F 32) 1 Fahrenheit degree = 5/9 Celsius degrees (for intervals) Force * 1 pound = 1 lb = N Pressure 1 pound per square inch =1 psi = kPa 1 standard atmosphere =1 atm. = kpa Energy, Work 1 British Thermal Unit =1 BTU = J 1 foot pound force =1 ft-lb = J 1 calorie =1 cal = J 1 kilowatt hour =1 kw h = 3.6 MJ Power 1 horsepower (550 ft-lb/s) = W 1 horsepower =1 Hp = 746 W (Electric) * The pound mass is used for measuring common everyday quantities, as for example groceries. The pound force is used in science and engineering. When converting units, we must know what context the pound is to be converted. If pounds are to be changed to kg, then 1 lb kg. If pounds are to be changed to N, then 1 lb = N. 19

22 When converting units, we use the basic conversion factors between units. Also, we keep in mind that the ratio of 2 equivalent quantities is 1. 1inch FOR EXAMPLE Since 1 inch = cm, = 1, also 2.54 cm inch = 1 The following examples illustrate how to convert units: EXAMPLE 1 Convert 3.46 in to mm Solution 3.46 in = 3.46 in make an equation 25.4 mm 3.46 in = ( ) 3.46in multiply right side by the conversion 1in factor so that inches (in) cancel 3.46 in = mm perform calculation round off answer to the same number 3.46 in = 87.9 mm of significant digits as was given in the quantity to be converted EXAMPLE 2 Convert 1.34 ft to cm Solution 1.34 ft = 1.34 ft make an equation 12in multiply right side by 1.34 ft = ( ) 1.34 ft the conversion factor 1ft so that ft cancel multiply right side by 12in 25.4 mm 1.34 ft = ( ) the conversion factor 1.34 ft 1ft 1in so that inches (in) cancel 12in 25.4 mm 1cm multiply right side by ( ) 1.34 ft = 1.34 ft the conversion factor 1ft 1in 10mm so that mm cancel 1.34 ft = cm perform calculation 1.34 ft = 40.8 cm round off answer With a little practice, the above steps can be combined in a single step as shown: 1.34 ft = 12in 25.4 mm 1cm ( 1.34 ft) 1ft 1in 10mm The measurement to be converted is multiplied by conversion factors until the needed units are left. The calculation is then performed and the answer is rounded off to the same number of significant digits as is given in the quantity to be converted. 20

23 EXAMPLE 3 Solution Convert 36.5 mi/min to km/hr 36.5mi km 60min 36.5 mi/min = min 1mi 1hr = 3520 km/hr EXAMPLE 4 Solution EXAMPLE 5 Solution EXAMPLE 6 Solution Here it is useful to know that 1 mi = km Convert 24.6 yd to m, knowing only that 1 yd = 36 in, l in = 25.4 mm and 1000 mm =1 m 36in 25.4 mm 1m 24.6 yd = 24.6 yd = 22.5 m 1yd 1in 1000 mm ( ) Convert 348 lb/ft 2 to N/m N ( ) 348lb/ft 2 1ft 1ft 1lb m m 348 lb/ft 2 = = N/m 2 NOTE: Here we had to convert ft to m twice because of the ft 2 units Convert 136 gal to kl 4.546l 1kl 136 gal = 136 = kl ( gal) 1gal 1000l EXAMPLE 7 Convert 12.5 ºF to ºC Solution Here we must use the formula 5 C = ( ) = therefore 12.5 ºF = 10.8 ºC 5 C = ( F 32) to convert 9 21

24 EXAMPLE 8 Solution The temperature difference between the inside and the outside of a wall is 45 ºF. What is the temperature difference in ºC Here, it is required to find the number of Celsius degrees that 45 Fahrenheit degrees are. The conversion factor to use for intervals is: 5 1 F = C F = ( 45 F) C /1 F 9 5 = 45 C 9 = 25 ºC Therefore, the temperature difference is 25ºC 22

25 Experiential Activity Four 1. Convert the following: a) 324 in to m b) 624 ft to km c) in to mm d) 658 cm to yd e) yd to mm f) 63.7 km to mi 2. Change the following: a) 157 in 2 to m 2 b) 257 ft 3 to m 3 c) 456 acres to ha e) m 3 to ft 3 f) mi 2 to km 2 g) 624 cm 2 to in 2 3. Convert the following: a) 3500 lb/in 2 to N/m 2 b) 380 ft/s to km/h c) 3.5 hp to kw d) 271 lb/ft 3 to N/m 3 e) 2370 ft-lb to J f) 275 lb/ft 2 to Pa g) 275 N/m 2 to lb/ft 2 h) 875 J to ft-lb i) 456 l/s to gal/min j) Btu/h to kj/s Show Me. k) 125 ºF to ºC Experiential Activity Four Answers 1. a) 8.23 b) c) d) 7.20 e ) 2995 f) a) b) 7.28 c) 185 d) 4.77 e) f) a) b) 417 c) 2.61 d) e) 3210 f) g) 5.74 h) 645 i) 6020 j) 36.7 k)

26 Practical Application Activity Complete the Units of Measurement and Unit Analysis assignment in TLM. Summary This module familiarized the student with the metric system, the Imperial system, and conversions between the two systems. 24

27

28