2.5. DIMENSIONAL ANALYSIS |
71 |
2.5Dimensional analysis
An interesting parallel to the “unity fraction” unit conversion technique is something referred to in physics as dimensional analysis. Performing dimensional analysis on a physics formula means to set it up with units of measurement in place of variables, to see how units cancel and combine to form the appropriate unit(s) of measurement for the result.
For example, let’s take the familiar power formula used to calculate power in a simple DC electric circuit:
P = IV
Where,
P = Power (watts)
I = Current (amperes)
V = Voltage (volts)
Each of the units of measurement in the above formula (watt, ampere, volt) are actually comprised of more fundamental physical units. One watt of power is one joule of energy transferred per second. One ampere of current is one coulomb of electric charge moving by per second. One volt of potential is one joule of energy per coulomb of electric charge. When we write the equation showing these units in their proper orientations, we see that the result (power in watts, or joules per second) actually does agree with the units for amperes and volts because the unit of electric charge (coulombs) cancels out. In dimensional analysis we customarily distinguish unit symbols from variables by using non-italicized letters and surrounding each one with square brackets:
|
|
|
|
|
|
|
P = IV |
|
|
|
|
|
|
|
|
|
|
|
[Watts] = [Amperes] × [Volts] |
|
or |
[W] = [A][V] |
|
|
|
||||||||||
|
Joules |
|
|
Coulombs |
|
|
Joules |
|
|
J |
|
|
C |
|
J |
|
|
|
|
= |
|
|
|
× |
|
|
or |
|
|
|
= |
|
|
|
|
Seconds |
Seconds |
Coulombs |
s |
s |
C |
||||||||||||
Dimensional analysis gives us a way to “check our work” when setting up new formulae for physicsand chemistry-type problems.
72 |
CHAPTER 2. PHYSICS |
2.6The International System of Units
The very purpose of physics is to quantitatively describe and explain the physical world in as few terms as possible. This principle extends to units of measurement as well, which is why we usually find di erent units used in science actually defined in terms of more fundamental units. The watt, for example, is one joule of energy transferred per second of time. The joule, in turn, is defined in terms of three base units, the kilogram, the meter, and the second:
[J ] =
[kg][m2]
[s2]
Within the metric system of measurements, an international standard exists for which units are considered fundamental and which are considered “derived” from the fundamental units. The modern standard is called SI, which stands for Syst`eme International. This standard recognizes seven fundamental, or base units, from which all others are derived4:
Physical quantity |
SI unit |
SI symbol |
|
|
|
Length |
meter |
m |
Mass |
kilogram |
kg |
|
|
|
Time |
second |
s |
Electric current |
ampere |
A |
|
|
|
Temperature |
kelvin |
K |
Amount of substance |
mole |
mol |
|
|
|
Luminous intensity |
candela |
cd |
An older standard existed for base units, in which the centimeter, gram, and second comprised the first three base units. This standard is referred to as the cgs system, in contrast to the SI system5. You will still encounter some derived cgs units used in instrumentation, including the poise and the stokes (both used to express fluid viscosity). Then of course we have the British engineering system which uses such wonderful6 units as feet, pounds, and (thankfully) seconds. Despite the fact that the majority of the world uses the metric (SI) system for weights and measures, the British system is sometimes referred to as the Customary system.
4The only exception to this rule being units of measurement for angles, over which there has not yet been full agreement whether the unit of the radian (and its solid counterpart, the steradian) is a base unit or a derived unit.
5The older name for the SI system was “MKS,” representing meters, kilograms, and seconds. 6I’m noting my sarcasm here, just in case you are immune to my odd sense of humor.