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SI Units

Learning Outcome When you complete this module you will be able to:

Perform simple calculations involving SI units.

Learning Objectives Here is what you will be able to do when you complete each objective:

1. List basic SI units and their symbols. 2. Identify and list symbols for unit prefixes. 3. Perform unit analysis in simple problems. 4. List derived SI units and their symbols. 5. Perform conversions between SI and Imperial units.

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INTRODUCTION Various systems of measurement have been used in different parts of the world for centuries. However, because each system was designed with its own base units, conversions from one system to another presented many problems. It became obvious that a standard system would have to be developed. This system would have to be precise, to allow for accurate measurement, yet simple enough to allow for conversions from one unit to another within the system. Such a system was developed in France in the seventeenth century, and was gradually adopted by other countries. This system was called the Metric System. In 1960, the latest version of the Metric System was developed, and named Le Systeme International d’unites (the International System of Units), more commonly called SI. BASIC UNITS There are seven base units in SI, as shown in Table 1:

Quantity Name of Base Unit Symbol

Length Mass Time Electric current Thermodynamic temperature Amount of substance Luminous intensity

metre

kilogram

second

ampere

kelvin

mole

candela

m kg s A K mol cd

Table 1

SI Base Units Note: The base unit kilogram is the only base unit with a prefix. The kilogram

was selected as a base unit since the gram was considered to be too small to be functional.

Several SI base units may be familiar to you, and others may not. In any case, you must know this chart thoroughly before proceeding with this module.

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Multiples and Submultiples of Base Units One area in which SI is simple to work with, compared to other systems, is that of converting small units to large, and vice versa. For example, suppose we wish to convert a distance, given in Imperial units, to a value with smaller units. Let’s do a conversion from miles to inches. First, the distance in miles would be converted to yards, or feet, using 1 mile = 1760 yards, or 1 mile = 5280 feet. Then, using 1 foot = 12 inches, or possibly 1 yard = 3 feet, we could arrive at the correct value and units. However, the conversion factors have to be exact, and an error could occur simply because the conversion may require several steps to complete. In SI, conversion factors are not required. Changing from a larger unit to a smaller one, or vice versa, requires only that you multiply or divide the unit by 10 or a multiple of 10 (i.e., 100, 1000, etc.) This can be done simply by moving a decimal point, in most cases. A specific set of prefixes is used to denote the resulting units after a decimal point has been moved. Table 2 indicates the various prefixes, and the asterisks indicate those most commonly used.

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Prefix Symbol Factor by Which Unit is Multiplied

* * * * *

exa peta tera giga mega kilo hecto deca deci centi milli micro nano pico femto atto

E

P

T

G

M k h

da d c

m

µ n p f a

1 000 000 000 000 000 000 1 000 000 000 000 000 1 000 000 000 000 1 000 000 000 1 000 000 1 000 100 10 1 0.1 0.01 0.001 0.000 001 0.000 000 001 0.000 000 000 001 0.000 000 000 000 001 0.000 000 000 000 000 001

1018 1015 1012 109 106

103

102

101 100 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18

Table 2

Metric Prefixes

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To illustrate the use of prefixes, let’s consider the unit of length in SI, the metre, with some prefixes.

Name Symbol Meaning Multiply Metres By

megametre kilometre hectometre decametre decimetre centimetre millimetre micrometre

Mm

km

hm

dam

dm

cm

mm

µm

one million metres one thousand metres one hundred metres ten metres one tenth of a metre one hundredth of a metre one thousandths of a metre one millionth of a metre

1 000 000 1 000 100 10 0.1 0.01 0.001 0.000 001

Table 3

Metric Prefixes Although any multiple or submultiple of a unit may be used, it is recommended that prefixes representing 10 raised to the power of a multiple of 3 (i.e. 10-3, 103, 10-6, 106, 109, etc.) are selected. For example, kilometre (km = 103 x m) is preferred to hectometre (102 x m). The following are examples of conversions from one prefix to another: Example 1: Convert 0.723 km to metres. Solution: 1 km = 1000 m, or there are one thousand metres per kilometre, written as 1000 m/km. For calculation purposes, we could write 1000 m/km as 1000 m

km

(The reason for this is explained shortly.)

Then: 0.723 km = 0.723 km × 1000 mkm

= 723 m (Ans.)

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Example 2: Convert 0.045 m to millimetres. Solution: Since 1 m = 1000 mm 0.045 m = 0.045 m x 1000 mm

m = 45 mm (Ans.) Example 3: Convert 109 mm to centimetres. Solution: Since 1 cm = 10 mm 109 mm = 109 mm x 1 cm

10 mm

= 10.9 cm (Ans.) Unit Analysis Notice that in the above examples, the units are included in the solutions. Units should be included in all calculations if possible. If you conduct a "unit analysis" for a solution, you will find that when the units for the final solution are proven to be correct, then the numeric result is usually correct. This is assuming, of course, that you have not made any math errors. For example, a unit analysis for Example 3 would be:

mm × cmmm = cm

The millimetres cancel out, leaving only cm in the final answer. mm– – × cm

mm– – = cm If you had solved the problem this way: 109 mm × 10 mm

1 cm= 1090 cm

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A unit analysis would reveal your error:

mm × mmcm = mm2

cm , which is not a unit of length For simple problems, such as Example 3, do a unit analysis at the same time as you solve the problem: 109 mm × 1 cm

10 mm= 10.9 cm

For more advanced calculations, you may find it easier to perform a separate unit analysis. The important thing to remember is: Always do a unit analysis. Example 4: Find the area of a rectangle that is 1.5 m long, and 93 cm wide. Use the formula, A = L x W. In cases like this, where we are going to multiply units together, we will have to change one, so that both units are the same. In this example, the final unit was not specified, so there are various ways of solving the problem. Solution 1: Change both dimensions to centimetres. A = L x W = (1.5 m x 100 cm/1 m) x 93 cm = 150 cm x 93 cm = 13 950 cm2 (Ans.) Solution 2: Change both dimensions to metres. A = L x W = 1.5 m x (93 cm x 1 m/100 cm) = 1.5 m x 0.93 m = 1.395 m2 (Ans.)

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Alternate Solution 2: From Solution 1: A = 13 950 cm2

Since: 1 m2 = 1 m x 1 m = 100 cm x 100 cm = 10 000 cm2

Then: 1.395 m2 = 1.395 m2 x (10 000 cm2/1 m2) = 13 950 cm2 (Ans.) Example 5: Convert 278 827 cg to kg. Solution: 278 827 cg x (1 g/100 cg) x (1 kg/1000 g) = 2.788 27 kg (Ans.) Example 6: Convert 1 000 000 mm to km. Solution: 1 000 000 mm x (1 m/1000 mm) x (1 km/1000 m) = 1 km (Ans.) Notice in these examples that the conversion was done in two steps. Sometimes it is easier to convert a value to unity (1 kg, 1 m, etc.) and then to convert to the final unit.

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Writing SI Symbols You should now be ready to attempt some problems using SI units and symbols. However, before you do, learn the following basic rules for writing SI symbols and for writing numbers with SI units. These rules must always be followed: 1. Always use the correct symbol. For example, use kg for kilogram, rather than

something like klg, which does not exist. 2. Symbols must always be written vertically. For example, use kg, not kg. 3. Symbols are lower case letters, unless they are derived from a proper name.

For example, use s (second), not S. (One exception to this rule will be mentioned later.)

4. Symbols are always singular. For example, 75 kg, not 75 kgs. 5. Do not use a period after a symbol, unless the symbol is at the end of a

sentence. For example, use "...27 m long.", not "...27 m. long." 6. Do not leave a space between a prefix and a unit symbol. For example, use

kg, not k g. 7. Use only the symbol with numerals. Use the full name of the unit when the

number is written out. For example: Use 5 m, not 5 metres. Use five metres, not five m.

8. Leave a space between a number and a symbol. For example, use 18 kg, not

18kg. An exception to this would be when a letter does not follow a number, such as in 18°C.

9. When writing numbers, use decimals rather than fractions. For example, use

1.5 kg, not 1 1/2 kg. 10. If a number is less than one, use a zero before the decimal point. For

example, use 0.8 m, not .8 m. 11. Numbers must be separated into blocks of three digits each, instead of using

commas. For example, use 78 232 456.738 92 mm, not 78,232,456.73892 mm. However, if a number has only four digits, a space is optional. For example, use 3.1416 cm or 3.141 6 cm.

12. When multiplying, use the multiplication sign (x) instead of a dot. For

example, use 178 x 97.3 m, not 178 • 97.3 m.

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MEASURING PHYSICAL QUANTITIES The following is an introduction to several other units that are used in the SI system. The topics are discussed in greater detail in other modules. Length Although the SI unit of length is the metre (m), multiples or submultiples are often used in everyday situations. Engineering drawings frequently use millimetres (mm), the textile industry often uses centimetres (cm), and highway distances may be given in kilometres (km). For navigational purposes, the nautical mile has been in use for a long time, and is presently included in SI. A nautical mile is a distance of 1852 m. Area In SI, the product of any two quantities produces the unit of the resultant quantity. This simply means, for example, that unit length (1 m) multiplied by unit width (1 m) equals unit area (1 m2). Although the square metre (m2) is the unit of area, the square centimetre (cm2) and the square millimetre (mm2) are often used. When measuring land area, such as a farm, the hectare (ha) is often used. 1 ha = 1 hm2 = 1 hm x 1 hm = 100 m x 100 m = 10 000 m2 For measuring extremely large geographical areas, the square kilometre (km2) is used. Volume The cubic metre (m3) is the unit of volume in SI. 1 m3 = 1 m x 1 m x 1 m The cubic centimetre (cm3) is often used for laboratory work. 1 cm3 = 1 cm x 1 cm x 1 cm The cubic decimetre (dm3) is also used for measuring solids, liquids, or gases.

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1 dm3 = 1 dm x 1 dm x 1 dm = 0.1 m x 0.1 m x 0.1 m = 0.001 m3 The cubic decimetre is given the name "litre". The symbol for the litre is L. (If you refer back to the rules on writing of SI symbols, the litre (L) is the one exception referred to in Rule 3).

Since, 1 dm3 = 1 L

and, 1 L = 0.001 m3

then, 1000 L = 1 m3 This fact can be useful when dealing with problems, and you should become very familiar with the litre (L). The millilitre (mL) is often used as a unit of volume in medicine, cooking, and laboratories. Speed Speed is defined as the distance a body travels in a unit of time. (Speed and velocity are not indentical, although their units are.) The unit for distance is the metre (m), and the unit for time is the second (s), as we already know. In SI, the quotient of any two quantities produces the unit of the resultant quantity. In the case of speed, a distance of one metre (1 m) divided by a time of one second (1 s) results in a unit of speed of one metre per second (1 m/s). The kilometre per hour (km/h) is often used to describe automobile and airplane speeds.

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Ships and aircraft often use the knot (kn) as a measure of speed. One knot is a speed of one nautical mile per hour. 1 kn = one nautical mile per hour = one thousand eight hundred and fifty-two metres per hour

(1852 m/h) Since,

1 hr = 3600 s (60 s/min x 60 min/h)

Then,

1 kn = 1852 m/h x (1 h/3600 s) = 0.514 m/s (approximately) Hopefully, you paid particular attention to the manner in which the unit for speed (m/s) was written. When two symbols are combined to form a unit, the following rules must always be applied: 1. Use a slash (oblique stroke) with symbols rather than the word "per". For

example, use km/h, not km per h. 2. Use the word "per" when writing full names of symbols rather than a slash.

For example, use kilometres per hour, not kilometres/hour. Acceleration Acceleration is defined as the rate of change of velocity. You have learned that the unit of velocity is m/s. The word "rate" indicates a unit of time, the second (s). So acceleration deals with change in velocity (m/s) per unit of time, the second (s), or acceleration is metres per second, per second. The unit becomes m/s/s. Then, metres per second, per second = m/s

s = (multiply top and bottom by 1/s, which does

not change the value). = m/s × 1/s

s × 1/s

= m/s × 1/s

1

= m/s2 The unit for acceleration then is m/s2.

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Mass The mass of a body refers to the quantity of matter that it contains. (The term "weight" is not the same as mass, and the difference between the terms is explained in another module.) In everyday use, the term "weight" is frequently misused to mean "mass" when working in SI. The unit of mass is the kilogram (kg). Large masses may be expressed in megagrams (Mg), equal to one thousand kilograms. This is also called a metric ton or tonne (t). 1000 g = 1 kg 1000 kg = 1 Mg = 1 t Density Density refers to the mass of a unit volume of a substance. Since mass is in kilograms (kg), and volume is in cubic metres (m3), it is obvious that the unit of density is kilograms per cubic metre, kg/m3. Temperature This topic is discussed in detail in another module. For now, let’s just state that the degree Celsius (°C) is the unit encountered in everyday use. For scientific work, the unit of temperature is the kelvin (K). The terms "Celsius degree", "degree Kelvin", "°K", and "deg" are all incorrect, and must not be used when expressing a temperature. Force Force is covered in detail in another module, so the definition will not be given here. However, part of the definition of force involves the idea of imparting acceleration (m/s2) to a mass (kg). The force required is calculated using the following formula: Force = mass x acceleration = kg x m/s2 = kgm/s2 The unit kgm/s2 is called a newton (N). This unit is very important, because it is used to form several other units as well.

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Work Work is done when a force causes a body to move through a distance. The following formula can be used: Work = force x distance = N x m = Nm (newton metre) The newton metre (Nm) is called a Joule (J). Energy The term energy means the ability to do work. In other words, energy can be converted to work. Since the unit of work is the joule (J), then energy must also be expressed in joules, or a multiple of joules. There are many forms of energy, and you will discover that various formulae are used to calculate various forms of energy. As you encounter each form of energy, do a unit analysis for each formula, to prove that the units are in joules or a multiple of joules. Power Power is defined as the rate of doing work. As was mentioned earlier, the word "rate" indicates a unit of time, the second (s). Power, then, is a measure of work, in joules (J), over a given time period in seconds(s). Power = work/time = J/s When one joule of work is done per second, we say the power developed is one watt (W). J/s = W The kilowatt (kW) and megawatt (MW) are usually used to indicate larger values of power.

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Pressure Pressure is defined as the force exerted over a unit area of a surface. Pressure = force/area = N/m2 A force of 1 N acting on an area of 1 m2 is called a pascal (Pa) Most equipment such as boilers, pumps, and compressors develop pressures which are expressed in kilopascals (kPa) or megapascals (MPa), because the pascal (Pa) is a relatively low pressure. A unit of pressure sometimes associated with SI is the bar. 1 bar = 100 kPa However, the bar is not a recognized unit in SI. A millibar (mbar) is equal to a pressure of one hundred pascals (100 Pa). It may be used, but only when performing international meteorological work. Note: You probably have noticed that some of the units introduced in the last

few topics have been given a symbol beginning with a capital letter. These symbols are derived from a proper name. Some examples are newton (N), joule (J), watt (W), and pascal (Pa). Refer to the section on "Writing SI Symbols", Rule 3.

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Table showing commonly used units.

Quantity Unit Symbol Derivation

Area Volume Speed Acceleration Density Specific volume Frequency Force Pressure Energy (work) Power Concentration Molar mass Molar volume Mass flow rate

square metre cubic metre metre per second metre per second squared kilogram per cubic metre cubic metres per kilogram hertz newton pascal joule watt mole per litre gram per mole cubic metre per mole kilogram per second

m2 m3 m/s m/s2 kg/m3 m3/kg Hz N Pa J W mol/L g/mol (kg/kmol) m3/mol kg/s

m x m m x m x m distance per unit time change in speed per unit time mass per unit volume volume of a unit mass l/s; reciprocal time kg•m/s2; mass times acceleration N/m2; force per area N•m; force times displacement J/s; energy per time mole per unit volume mass per unit mole volume per unit mole kilogram per unit time

Table 4 Derived Units

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UNIT CONVERSIONS There may be occasions when units from one system may have to be converted to units of another system. The conversion chart shown below can be used to convert from SI units to Imperial units, and vice versa. Length 1 in = 2.54 cm 1 cm = 0.3937 in 1 ft = 0.3048 m 1 m = 3.28 ft 1 mile = 1609 m Area 1 in2 = 6.45 cm2 1 cm2 = 0.155 in2 1 ft2 = 0.093 m2 1 m2 = 10.75 ft2 1 sq. mile = 2.59 km2 1 km2 = 0.386 sq mile Volume 1 in3 = 16.39 cm3 1 cm3 = 0.061 in3 1 ft3 = 0.0283 m3 1 m3 = 35.336 ft3 Capacity 1 qt = 1.136 L 1 L = 0.88 qt 1 gal = 4.546 L 1 L = 0.22 gal Mass 1 lb = 0.454 kg 1 kg = 2.2 lb Force 1 lb = 4.448 N 1 N = 0.225 lb Pressure 1 lb/in2(psi) = 6.895 kPa 1 kPa = 0.145 lb/in2(psi) 1 bar = 100 kPa 1 bar = 14.51 psi 1 psi = 0.069 bar

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Energy 1 ft lb = 1.356 J 1 J = 0.737 ft lb 1 Btu = 1.055 kJ 1 kJ = 0.948 Btu 1 kcal = 3.968 Btu 1 Btu = 0.252 kcal 1 kcal = 4.186 kJ 1 kJ = 0.239 kcal 1 hp-hr = 2.685 MJ 1 MJ = 0.372 hp-hr 1 watt-hr = 3.6 kJ 1 kJ = 0.278 watt-hr Power 1 hp = 0.746 kW 1 kW = 1.34 hp Example 7: Convert 22 miles to kilometres. Solution: From the chart, 1 mile = 1609 m = 1.609 km 22 miles = 22 miles x 1.609 km/1 mile = 35.398 km (Ans.) Example 8: Convert 13 790 kPa to pounds per square inch. Solution: From the chart, 1 psi = 6.895 kPa 13 790 kPa = 13 790 kPa x 1 psi/6.895 kPa = 2000 psi (Ans.)

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As you become familiar with SI, some of the commonly used conversions will become familiar to you. For example, you will probably remember that 1 inch = 2.54 cm, or 25.4 mm. Another common conversion is 1 psi = 6.895 kPa. Once you have memorized some of the common conversions, you should be able to derive some of the others. For example, suppose a cube has sides of 3.75 feet, and you wish to calculate the number of square metres in the cube. You could solve the problem this way, without referring to a conversion chart, knowing from memory only that 1 inch = 2.54 cm: Each side length = 3.75 feet x 12 inches/1 foot = 45 in Side length in SI = 2.54 cm/inch x 45 inches = 114.3 cm 114.3 cm x 1 m/100 cm = 1.143 m Volume of cube = (length of side) 3 = (1.143 m) 3 (Notice that the unit, m, is also cubed.) = 1.49 m3 (Ans.) (Rounded off to 3 significant digits). As you become more proficient with conversions, you will probably do the same problem more quickly, such as this: Side length = (3.75 ft x 12 in/ft) x 2.54 cm/in x 1 m/100 cm = 1.143 m Volume = (1.143 m) 3 = 1.49 m3 (Ans.)

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Self Test After completion of the self test, check your answers with the answer guide that follows. 1. Name the SI base units for the following, and in brackets behind each one,

write its symbol: (a) length (b) mass (c) time. 2. Write the symbol for each of the following: (a) millimetre (b) megagram (c) decisecond (d) microgram (e) decametre (f) centimetre (g) millisecond (h) kilometre. 3. Convert the following to metres: (a) 289 cm (b) 1828 mm (c) 1.45 km (d) 17 dm (e) 17 dam. 4. Convert the following to centimetres: (a) 131 m (b) 2.7 km (c) 14 dm (d) 25.4 mm (e) 118.3 dam. 5. Convert the following to kilograms: (a) 450 g (b) 7.3 Mg (c) 3921.2 mg (d)

145 hg (e) 10 000 dg.

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Self Test Answers 1. (a) metre (m) (b) kilogram (kg) (c) second (s) 2. (a) mm (b) Mg (c) ds (d) µ g (e) dam (f) cm (g) ms (h) km 3. (a) 289 cm x 1 m/100 cm = 2.89 m (b) 1828 mm x 1 m/1000 mm = 1.828 m (c) 1.45 km x 1000 m/1 km = 1450 m (d) 17 dm x 1 m/10 dm = 1.7 m (e) 17 dam x 10 m/1 dam = 170 m 4. (a) 131 m x 100 cm/1 m = 13 100 cm (b) 2.7 km x 1000 m/km x 100 cm/m = 270 000 cm (c) 14 dm x 1 m/10 dm x 100 cm/1 m = 140 cm (d) 25.4 mm x 1 cm/10 mm = 2.54 cm (e) 118.3 dam x 10 m/dam x 100 cm/1 m = 118 300 cm 5. (a) 450 g x 1 kg/1000 g = 0.45 kg (b) 7.3 Mg x 1000 kg/1 Mg = 7300 kg (c) 3921.2 mg x 1 g/1000 mg x 1 kg/1000 g = 0.003 921 2 kg (d) 145 hg x 100 g/hg x 1 kg/1000 g = 14.5 kg (e) 10 000 dg x 1 g/10 dg x 1 kg/1000 g = 1 kg

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Notes: