Units and Dimensions

Units are part of physical quantities. This implies that a physical variable will have a value and may include a dimension such as length, time, mass or combinations of them. For example, x = 5 meters, t = 3 seconds, v = 5/3 meter per second = 5/3 m/sec, m = 10 kg or E = 2 J, etc.

Some variables can also be dimensionless, in which case they'll be just a number. Ratios between similar variables (e.g., between two lengths) will be dimensionless. Such numbers only have a size but no dimension. Physicists generally like dimensionless numbers because they are usually more meaningfull (e.g., a length of something relative to something else), and of course, their value does not depend on the unit system used!

Useful suggestions:
• Always check your units each step of a calculation! You'll be amazed to see how effective it is in finding many of your calculation errors. It won't find missing factors of 2, but will be probably find most other mistakes.

• Always check that your answers make sense. Following our previous example, we could have found out that we have an error in an additional way. We expect the time of fall of an object to decrease with increasing gravity, thus, the power of g in the expression has to be negative (though we cannot find the actual power this way).

No other suggestion will increase your GPA as much!
Example: Unit checking
Suppose you calculate the time it takes an object to fall from a height h, and you obtain $ t = \sqrt{2 h g} $ as the result. Is this answer reasonable?

To see this, let's plug in the units. In M.K.S., h has units of meters, and g of m/sec2. Plugging in, we find $ [t] = [\sqrt{2 h g}] = \sqrt{m \times m/s^2} = m/sec \neq sec $ which has units of velocity, not of time! The correct units are obtained for $ t = \sqrt{2 h/g} $. Thus, not only can be find that we have an error, we can also guess how to correct it (in this case, we accidentally multiplied by g instead of dividing by it). Of course, we have no way of knowing this way whether the factor of 2 is correct. So we cannot find this way all the errors possible...
Useful note:
Some students tend to forget writing the units when they're needed (e.g., write x=5 instead of x=5m). Others implicitly write the units twice. e.g. write x1=2 x2 [m]. This is wrong because x2 already has the units in it, such that the r.h.s. has units of length squared! In other words, don't forget the units when the are required, but don't be over zealous to put them when they are not required!

Unit Systems

The same physical quantities which have dimensions will have different values when working with different basic dimensions. For example, 2m is 200cm, and it is also 6.56167979 feet. Similarly, one can work with gr, kg, lb, ton, etc, or choose yrs or fortnights instead of seconds. The set of basic units one chooses to work with is called a unit system.

The most common unit systems in science are the M.K.S. which includes the m, kg and sec. The second system is the c.g.s., which uses cm, gr and second. Another common system is the English system (lb, ft, sec), which was rightfully abandond even by the English themselves (through mind you, the moon was reached using the English system!).

Which system is best? Different people prefer working with different systems. In some cases, there is some logic to prefer one over the other. In other cases, there is none. For example, once you will study electricity and magnetism, you will realize that the E&M related units in c.g.s. are significantly better than their equivalent M.K.S. counter parts. If you are trying to solve a problem in astrophysics, you might prefer using the solar mass as a unit mass instead of a kg!

The bottomline, you should be able to work with any unit system you wish. In astrophysics, c.g.s. is very common (which is why I prefer it), but in other branches of physics, the SI (system international = M.K.S.) dominates. So in these notes, you will encounter both.

Useful trick:
You can use google to convert units. For example, type in google "5 ft in m" or "17 erg in J". It even works with archane units, such as "furlong/fortnight in cm/min", or more complex ones such as "0.7 tesla in gauss". Just a reminder though, in the exam you wont have google to convert units for you. ;-)