Everything You Wanted to Know About Units, but were Afraid to Ask
Units of Measurement
Scientists use numbers (or measurements) to help describe data because numbers don’t lie
(People do, but numbers don’t). When they use numbers it is helpful if the person looking at the
numbers has some idea about what the numbers mean. The measurement of any quantity is made
relative to a particular standard or unit, and this unit must be specified along with the numerical value of
the quantity. So scientists use units of measurement to describe what these numbers represent. It
would be real confusing if I said, “Give me 15.” What do I mean ? We could guess, but we really
wouldn’t be sure. Do I mean 15 minutes, $15, or slap my hand 3 times ? If I had used an appropriate
unit the confusion would have been cleared up.
AP Physics Class Rule
If the problem has units, so does your answer.
It is important to have standards for your units of measurement so that 1 foot for one person is
the same as one foot for another. If this weren’t true, life could get very complicated. In medieval times
this was a problem. In some kingdoms the foot was defined as the length of the king’s foot. If two
kings from different kingdoms had different size feet, a foot long hot dog for one subject would be
shorter than for a subject in the other kingdom. Even worse, when that king died and his son assumed
the throne, the foot would be redefined, once again causing confusion. Finally it was decided that there
should be standard units for everyone, and this is what we have today.
The first real international standard was the meter (m) established as the standard of length by
the French Academy of Sciences in the 1790’s. In a spirit of rationality, the standard meter was
originally chosen to be one ten millionth of the distance from the Earth’s equator to either pole, and a
platinum rod was made to represent this length. Later, as technology improved, the meter was redefined
to provide greater precision. Now the standard unit of length, the meter (m), is defined as the distance
light travels in a vacuum during a time interval of 1/299,792,458 of a second. The standard unit of time,
the second (s), is defined as time it takes for a cesium atom to decay 9,192,631,770 times. The standard
unit of mass, the kilogram (kg), is the mass of a particular cylinder made of a platinum-iridium alloy
which is stored at the International Bureau of Weights and Measures near Paris, France.
In the US we use British units. We measure length in inches, feet and miles. We measure time
in seconds, minutes and hours. We measure mass in ounces. pounds of mass and slugs (and salt won’t
kill these slugs) and we measure weight in pounds of force (not the same as pounds of mass). This
seems to work for us, however these units are defined in terms of the standard units. For example, 1
inch is defined as exactly 0.0254 m.
A vast majority of the world uses the metric system for their system of measurement. In the
metric system, the larger and smaller units are defined in multiples of 10 from the “base unit”. It turns
out that for consistency, the scientific community has agreed to use this system for measurement. We
call this system the International System of Units or the SI1 system.
1
Why not IS ? SI stands for Syste´me International which is French for International System.
Converting Units
Lets say we are measuring something in centimeters but I need the measurement in meters. To
get this answer I would need to convert my units from centimeters to meters by determining the
conversion factor for each unit and setting up a table to cancel the units until you have the units you
want. This is called factor labelling or dimensional analysis. Some examples are shown below.
Metric Conversion Factors
Conversion
Symbol
Prefix
Factor
exa
1018
E
P
peta
1015
T
tera
1012
G
giga
109
M
mega
106
k
kilo
103
d
deci
10-1
c
centi
10-2
m
milli
10-3
µ
micro
10-6
n
nano
10-9
p
pico
10-12
f
femto
10-15
Example 1
Convert 27-fm to m
The conversion factor for femto is 10-15. This means there are 10-15-m in 1-fm.
27-fm
10-15-m
1-fm
=
2.7 X 10-14-m
.
Example 2
Convert 335-s to Ms
The conversion factor for Mega is 106. This means there are 106-s in 1-Ms.
335-s
1-Ms
106-s
=
3.35 X 10-4-Ms
Example 3
Convert 3.25 X 1012-nJ to EJ
Since this requires us to go from nJ to EJ, we will need two steps to do this. We will convert nJ to J and then convert J to EJ.
The conversion factor for nano (n) is 10 -9. This means there are 10-9-J in 1-nJ
The conversion factor for Exa (E) is 10 18. This means there are 1018-J in 1-EJ.
3.25 X 1012- nJ
10-9-J
1-nJ
1-EJ
1018-J
=
3.25 X 10-15-EJ
Sometimes the units we are converting have powers. This is true for units of area and for units
of volume. To convert these units we just raise the conversion factors to the power of the unit.
Example 4
Convert 75-cm2 to km2
Since this requires us to go from cm2 to km2, we will need two steps to do this. We will convert cm2 to m2 and then convert
m2 to km2.
The conversion factor for centi (c) is 10-2. This means there are (10-2)2-m2 in (1)2-cm2
The conversion factor for kilo (k) is 103. This means there are (103)2-m2 in (1)2-km2.
75-cm2
(10-2)2-m2
(1)2-cm2
(1)2-km2
(103)2-m2
=
7.5 X 10-9-km2
When working with speed and density, the units are fractions already. To convert these types of
units you just place the unit in the denominator into the bottom of your table when your start.
Example 5
Convert 100-m/s to km/hr
This will require us to do two conversions. We will need to convert from m to km and also from s to hr
The conversion factor for kilo (k) is 103. This means there are 102-m in 1-km
There are 3600-s in 1-hr. (60-s X 60-min)
100-m
1-s
1-km
103-m
3600-s
1-hr
=
360-km/hr
If you need to convert from English units to metric units, you need to know the conversion factors. The
table below gives some conversions you may need, of course any science text or the internet can give
you conversions not found here.
or Mass
𝟓
𝐂 = 𝟗(𝐅 − 𝟑𝟐)
Working with Units in Mathematical Formulas
Formulas are very important in Physics. We use formulas to describe the relationship between
physical quantities, and to calculate the value of other physical quantities. The numbers we substitute
into these formulas are usually measurements with units, so we need to be careful when we calculate
using units.
Adding or Subtracting
When we add or subtract, we need to be working with the same types of measurements. We say
the units are consistent. For example, we can add lengths together and time together, but we can’t add a
time to a length. If we are adding m to cm, we need to convert one or the other so that they are the same
unit. Lets look at some examples.
Example 6
Calculate
a)
23 cm + 45 cm = 68 cm
b)
23 cm - 12 cm = 11 cm
c)
35 ps + 47 TJ = ?
(These aren’t the same type of measurements, so we can’t add them.)
d)
100 cm + 5 m = 1 m + 5 m = 6m
( We had to convert cm to m since they weren’t consistent to start with.)
Multiplying or Dividing
To multiply or divide with units, you just do the algebra. It doesn’t matter if the measurements
are the same type or even if they are consistent. You just do the algebra.
Example 7
Calculate
a)
200 m X 10 m = 2000 (m)(m) = 2000 m2
b)
30 N X 4 s = 120 (N)(s) = 120 N∙s
It is better to use N∙s here rather than Ns since the “N” could be confused with a metric prefix.
km
c)
400 km ÷ 20 hr = 20 (km)÷(hr) = 20 hr = 20 kilometers per hour
Astronomical Units
Did you know that the Earth is 1.496 x 1011 m from the Sun. This is fairly far. Other planets
like Uranus are even farther (2.872 x 1012 m). This is also far and to write this distance many times
would be inconvenient and for most purposes not very useful. Scientists would like a more descriptive
and understandable way of talking about distances inside our Solar System. What they came up with
was a unit of distance called an Astronomical Unit.
When you are measuring a distance in astronomical units (abbreviated AU) you are really just
comparing that distance to the distance the Earth is from the Sun. For example, if two objects are 2
astronomical units (2 AU) apart then they are twice as far apart as the Earth is from the Sun. Using
astronomical units allows us to easily compare distances, especialy large distances.
Converting a distance to AU is actually quite simple, you just take that distance and divide it by
the distance the Earth is from the Sun. There is a catch though. Your units must be the same. For
instance, if you want to convert a distance in meters to AU you divide by the number of meters the Earth
is from the Sun. If the distance happens to be in miles, you divide by the number of miles the Earth is
from the Sun.
Example 8
How many astronomical units is Uranus from the Sun?
Above I stated that Uranus was 2.872 x 10 12 m from the Sun. so to
convert this to AU I will divide it by the Earth’s distance from
the Sun (1.496 x 1011 m).
2.872 1012 m
 19.20AU
1.496 1011 m
This means that Uranus is slightly more than 19 times as far from
the Sun as the Earth.
