Module 1: Units and Measurements

Michael Cowan and Kevin Lenton


All measurements of physical quantities involve a number and some unit of measurement. For instance, a length may be measured in metres, in feet, in light years (a very large unit, the distance light travels in a year), or in Angstrom units (a very small unit, about the size of a single atom) but it has to be measured in something. Just stating that something is 6.3 long, without specifying the unit of measurement, tells us almost nothing about the object: it could be anything from a molecule to a cluster of stars !
Physicists always specify the units; you should too.

Three kinds of measurement

There are three basic kinds of measurement, or dimensions, which we will deal with here. All physical quantities we consider will be either one of the three or some combination of them.

1) Length

The basic scientific unit of length is the metre (abbreviated m), originally chosen as $10^{-7}$ (1 / 10 000 000) of the distance from the equator to the North Pole, later the distance between two fine scratches on a special metal bar kept in Sevres, France, and now defined as the distance light travels in vacuum in 1 / 299 792 458 of a second. If you want an intuitive feel for how large 1 metre is, think of it as a bit more than a yard (which is 3 feet) or half the height of a very tall person (6' 6" = 2m).

Other common units related to the metre are

centimetre (cm) $10^{-2}$m or 1 /100 of a metre (about the width of a fingernail)

millimetre (mm) $10^{-3}$m or 1 / 1000 of a metre (about the thickness of a fingernail) kilometre (km) 103 or 1000 m (about 3/5 of a mile).

The prefixes micro - $10^{-6}$ milli - ($10^{-3}$}) centi - ($10^{-2}$) kilo- ($10^{3}$) and Mega- ($10^{6}$) are extremely common and useful.
Learn them ! There are others, but we will not need them for this course. 

2) Time

The basic unit of time is the second (s), now defined as the time it takes light to travel 299 792 458 m in vacuum.

To get a feel for the second, count 'thousand and one, thousand and two, thousand and three, fairly quickly, or better yet, watch the second hand or the flashing digits of a digital watch.

Other common time units are:

microsecond ($\mu s=10^{-6}s$)

millisecond ($ms=10^{-3}s$)

minute (min) = 60 s, hour (h) = 60 min, day (d) = 24 h, and year (y) = 365.25 d

3) Mass

The basic unit of mass is the kilogram (kg), defined as the mass of a particular metal cylinder kept in Sevres, France. Holding 1 kg in your hand is like lifting 2.2 pounds, or a litre of water.

Other common mass units are

gram (g) =${\frac {1}{1000}}kg=10^{-3}kg$(about the mass of a five- dollar bill)

milligram (mg) $=10^{-3}g=10^{-6}kg$

1 tonne = metric ton = $10^{3}$ kg or the equivalent to 2200 lb (mass of a 1m cube filled with water)

All the physical quantities we will need will have some combination of the above set of units.

Scientists use the Metric System, and so should you:
The International System of Units (SI)

[To the Metric System]

Most scientists use SI units, which were defined in a convention in Paris.
Check out the history of the SI units

Measurement	SI Unit	SI Unit Symbol	Historical Context
Length	meter	m	[check it out]
Time	second	s	[check it out]
Mass	kilgram	kg	[check it out]

Note that SI units are often not used consistently. For instance, chemists use the gram g as the base unit of mass, even though the gram is not the SI unit. It is just that grams are more practical to use in chemistry than kg.

Derived Units

Other quantities, called derived quantities, are defined in terms of the base quantities. For instance

Measurement	SI Unit	SI Unit Symbol
Area	square meter	$m^{2}$
Volume	meter cubed	$m^{3}$
Velocity/Speed	meter/second	$m/s$ or $ms^{-1}$

CONVERTING UNITS

[The most entertaining unit conversion video ever made!]

Often we must express a physical quantity in different units. For example, your road map gives the distance to your destination as 350 mi, but your car odometer is in km. You want to express 350 miles (350 mi) in km, without changing the actual distance you must travel. You know (or look up on google) that 1 mi = 1.609 km; for our purposes, let us say 1 mi =1.6 km is precise enough.

Consider the fraction $({\frac {1.6km}{1mi}})$

Since 1.6 km actually equals 1 mi, this fraction is equal to 1. We can multiply 350 mi by it without changing the distance. Thus
$350mi=350mi\times ({\frac {1.6km}{1mi}})=560km$

Since there were units of mi in both the numerator (top) and the denominator (bottom), the units of mi cancelled out, leaving us with the distance in km .

For each unit to be converted, construct a fraction equal to 1 which cancels out the unwanted unit and replaces it with the desired one. Multiply by that fraction.

Example

Convert 90 km/h (typical speed limit on a country road) to m/s.
We now have two units to convert, km to m and h to s. We know that 1000 m = 1 km, so that fraction is $({\frac {1000m}{1km}})$ For the times units, we know that 1 hr = 60 min and 1 min = 60 s.

We also note that the time unit to be converted is in the denominator (km per hr, not km x h). Thus

we make the fractions $({\frac {1hr}{60min}})({\frac {1min}{60s}})$ to cancel the hr and substitute s.

Finally we have

$90{\frac {km}{h}}=90{\frac {km}{h}}({\frac {1000m}{1km}})({\frac {1hr}{60min}})({\frac {1min}{60s}})({\frac {1.6km}{1mi}})=25{\frac {m}{s}}$

Some quantities contain repeated units. For example, an area is a length multiplied by a length. To change an area in $m^{2}$ to $cm^{2}$ we need to convert each of the two length units. Thus we would multiply by

$({\frac {100cm}{1m}})({\frac {100cm}{1m}})={\frac {100^{2}cm^{2}}{1m^{2}}}={\frac {10^{4}cm^{2}}{1m^{2}}}$

A volume contains three length units. To change a volume in $cm^{3}$ to $m^{3}$, multiply by

$({\frac {1m}{100cm}})({\frac {1m}{100cm}})({\frac {1m}{100cm}})={\frac {1m^{3}}{100\times 100\times 100cm^{3}}}={\frac {1m^{3}}{10^{6}cm^{3}}}$

To avoid mistakes in building these fractions, always know which of your units is the smaller one. It takes many small units to equal one large unit. For example, you know that 1 mm is smaller than 1 m, so that 1000 mm = 1 m, not the other way around

Resources

Video

[Easy Examples]

[Complicated Example]