Units are a valuable tool; careful attention to the units involved at each stage of a calculation can help you catch and fix mistakes. Much confusion can be avoided if you work with units as though they were symbols like those in algebra. For example:
(5 m) × (2 sec) = (5 × 2) × (m × sec) = 10 m sec.
The units in this example are meters times seconds, pronounced as `meter seconds' and written as `m sec'.
(10 m) ÷ (5 sec) = (10 ÷ 5) × (m ÷ sec) = 2 m/sec.
The units in this example are meters divided by seconds, pronounced as `meters per second' and written as `m/sec'; these are units of speed.
(15 m) ÷ (5 m) = (15 ÷ 5) × (m ÷ m) = 3.
In this example the units (meters) have canceled out, and the result has no units of any kind! This is what we call a `pure' number. It would be the same regardless what system of units were used.
(5 m) + (2 cm) = (5 m) + (0.02 m) = (5 + 0.02) m = 5.02 m.
Recall that a `cm', or centimeter, is one hundredth of a meter. So 2 cm = (2 ÷ 100) m = 0.02 m.
(5 m) + (2 sec) = ???
Meters and seconds are different kinds of quantities; one is a length, and the other is a time. As a rule, we can't convert a length to a time, or a time to a length, so there is no way to add these quantities. (In Relativity Theory this rule has exceptions -- in Relativity, we can convert between units of length and time. But that's an advanced topic.)
Astronomers use a mixture of units, and we often have to convert one to another. Converting between different units is easier if you remember to treat units like symbols; you simply replace the original unit with its equivalent in the unit desired, and do the necessary arithmetic. For example:
6 ft = 6 × (1 ft) = 6 × (0.3045 m) = (6 × 0.3045) m = 1.84 m.
165 lb = 165 × (1 lb) = 165 × (0.454 kg) = (165 × 0.454) kg = 75 kg.
43 yr = 43 × (1 yr) = 43 × (3.15 × 10^{7} sec) = (43 × 3.15 × 10^{7}) sec = 1.35 × 10^{9} sec.
What about converting the other way? Again, treating units as symbols simplifies the problem:
5 m = 5 × (1 m) = 5 × (1 ÷ 0.3045) ft = (5 ÷ 0.3045) ft = 16.4 ft.
120 kg = 120 × (1 kg) = 120 × (1 ÷ 0.454) lb = (120 ÷ 0.454) lb = 264 lb.
Here are the factors needed to convert between the different systems of units used in astronomy. Notice that there are many different length units:
Years | 1 yr = 3.15 × 10^{7} sec |
Astronomical Units | 1 AU = 1.496 × 10^{11} m |
Light Years | 1 ly = 9.461 × 10^{15} m |
Parsecs | 1 pc = 2.062 × 10^{5} AU = 3.086 × 10^{16} m |
Kiloparsecs | 1 kpc = 10^{3} pc = 3.086 × 10^{19} m |
Megaparsecs | 1 Mpc = 10^{6} pc = 3.086 × 10^{22} m |
Earth Masses | 1 M_{} = 5.967 × 10^{24} kg |
Solar Masses | 1 M_{} = 1.989 × 10^{30} kg |
Joshua E. Barnes
(barnes@ifa.hawaii.edu)
Last modified: August 23, 2006 http://www.ifa.hawaii.edu/~barnes/ast110_06/units.html |