Chapter 1. Units and unit conversions. Systems of measurement

Cy Vergara

Learning Objectives

By the end of this chapter, you will be able to:

List physical quantities and their corresponding SI units;
Convert units from one system of measurement to another;
Solve problems involving length, area, and volume.

Introduction

In industrial settings and in real life, we are exposed to a lot of information regarding the size of various physical quantities. This size can present itself in the form of a length, area, speed, etc.

To be able to interpret and make sense out of these quantities we need to use a standardized system of units. These units help us describe and compare physical quantities in a meaningful way.

The magnitude (size) of a physical quantity does not tell the whole story if it is not accompanied by a unit. For example, if we say that a crate is subjected to a horizontal pulling force of 120, we do not know the amount of force, since the information could mean: 120 N, 120 lbs., 120 MN, 120 kN, etc. Unless followed by a unit, the magnitude is not helpful in any calculations, conclusions, or interpretations.

Concepts

1. Systems of measurements

The main systems of measurement used in North America are the SI system and the imperial system (British or American system).

SI (International System of Measurement) is the latest form of the metric system. It is a convenient system because conversions between metric units can be done simply by moving the decimal place of a number. This is because the metric prefixes are powers of 10. For instance, there are 1000 (or 103) m in 1 km, 0.000001 (or 10-6) m in 1 µm, etc.

Prefixes help with expressing very large or very small quantities, without having to introduce new units. For instance, in the construction industry we would use metres or centimetres to express lengths, but in the road building industry we would use kilometres to express the same physical quantity – length.

Figure 1.1 shows some examples of prefixes and conversions from one unit of length to another. For instance, to convert from km to m we multiply by 1000, whereas converting from m to km would require dividing by 1000.

Figure 1.1 Prefixes. Conversions between units of length in the SI system of measurement

In the imperial system, the relationships are less simple. For instance, there are 12 inches in a foot, 3 feet in a yard, 1,760 yards in a mile, and so on.

Figure 1.2 Conversions between units of length in the imperial system

The SI system contains three main elements:

Base units
Derived units
Prefixes

The table below represents the seven basic units of the SI system.

Table 1. SI base units
Quantity	Unit Name	Unit Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Electric current	Ampere	a
Temperature	Kelvin	k
Amount of substance	Mole	mol
Luminous intensity	Candela	cd

Derived units are units created from a combination of two or more basic units.
The table below shows some examples of derived units.

Table 2. Derived units (examples)
Quantity	Unit Name	Unit Symbol
Force	Newton = kg times metre per squared second	N, kg·m/s²
Torque	Newton times metre	N·m
Work	Joule = Newton times metre	J
Velocity	Metre per second	m/s
Acceleration	Acceleration Metre per squared second	m/s²

SI prefixes are shown in the table below.

Table 3. Prefixes used in conjunction with the SI system of measurement
Prefix	Symbol	Value	Example Name	Example Symbol	Example Value
exa	E	10¹⁸	Exametre	Em	10¹⁸m
peta	P	10¹⁵	Petasecond	Ps	10¹⁵s
tera	T	10¹²	Terawatt	TW	10¹²W
giga	G	10⁹	Gigahertz	GHz	10⁹Hz
mega	M	10⁶	MegaNewton	MN	10⁶N
kilo	k	10³	Kilometre	km	10³m
hector	h	10²	Hectolitre	hL	10²L
deka	da	10¹	Dekagram	dag	10¹g
		10⁰(=1)
deci	d	10^-1	Decilitre	dL	10^-1L
centi	c	10^-2	centimetre	cm	10^-2m
milli	m	10^-3	Millimetre	mm	10^-3m
micro	µ	10^-6	Micrometre	µm	10^-6m
nano	n	10^-9	Nanogram	ng	10^-9g
pico	p	10^-12	Picosecond	ps	10^-12s
femto	f	10^-15	Femtometre	fm	10^-15m
atto	a	10^-18	Attosecond	as	10^-18s

2. Unit conversions

Every physics formula comes with attached matching units. Using different units than those indicated for any specific formula will result in an incorrect answer.

A lot of times measurements done in a real-life setting or information collected from studies, research projects, etc., will provide us with measurements of physical quantities that do not correspond to those required in the formulas. This creates the need to convert units to those that match a specific type of calculation.

Another need for converting units arises when it is necessary to use consistent units of measurement in situations where some quantities are expressed in SI units and others in Imperial units.

A conversion factor is a ratio or a fraction equal to 1. You can multiply any number by 1 without changing the original value. When you multiply a number by a conversion factor, you are simply multiplying it by one.

The conversion factor will contain either:

the new (desired) units in the denominator and the old units in the numerator:

[latex]{\small \text{OLD UNITS}} \times \frac{\text{NEW UNITS (DESIRED UNITS)}}{\text{OLD UNITS}} \rightarrow {\small \text{NEW UNITS (DESIRED UNITS)}}[/latex]

or

the new units in the denominator and the old units in the numerator:

[latex]{\textstyle \frac{1}{\text{OLD UNITS}}} \times {\textstyle \frac{\text{OLD UNITS}}{\text{NEW UNITS (DESIRED UNITS)}}} \rightarrow {\textstyle \frac{1}{\text{NEW UNITS (DESIRED UNITS)}}}[/latex]

The conversion factor is constructed so that the old units will cancel out and the resulting quantity will be expressed in the new (desired) units.

Sometimes we will need more than one conversion factor to obtain the desired units.