Dimensional Analysis  Part 2
Of oranges, nukes, and virgins,

2.1
Introduction:
For many years, physics education in American schools has placed
great emphasis on mathematics as a tool for modeling physical
interactions. Over this same period of time, American schools have
chosen to minimize the role of dimensional analysis in their physics
curriculum. In my opinion this is a serious mistake. Without
dimensional analysis, the student is left with little more than a
plethora of seemingly unrelated mathematical equations, and no
rational method for understanding the deeper relationships embodied
by these equations.
When properly taught, dimensional analysis should satisfy a twin set of goals.
It is not enough to unify physics. The teacher must also unify the students understanding of physics.
2.2
Constructing the Universe:
From the perspective of physical mathematics, the entire universe is
constructed from just four simple or fundamental quantities.
These are:
ALL OTHER UNITS of measurement are compound units, constructed from these four fundamental units. Sound unbelievable? Well it's true, and when the physics student masters this profound concept, never again will he/she wonder if a particular physics equation is correct or not. The student who masters dimensional analysis will be able to determine, with nothing more than pencil & paper, the truth of any physics equation. Have you ever wondered why: E = MC^{2} When dimensional analysis is mastered, it becomes obvious...
2.2.1
The fundamental units of measure:
Throughout this paper, we shall use the following symbols for the
fundamental units:
L = Length in meters. M = Mass in kilograms. T = Seconds. Q = Electric charge in coulombs.
2.2.2
The kinetic units of measure:
The mechanical (kinetic) units of measure form an ascending ladder,
starting at velocity, and ending with power. We shall cover each
unit of measure as a separate topic, and show how it is related to
the previous unit, and more importantly, how it is constructed from
the four fundamental units.
2.2.3
Velocity:
Velocity is the ratio of length to time, or if you prefer length
divided by time. In physics, its unit of measurement is "meters per
second". Other common units of velocity measurement are "miles per
hour", and "furlongs per fortnight" (useful when traveling by
horseback). The mathematical form of velocity is shown in Eq. 1.
Where:
V = Velocity in meters per second.
2.2.4
Acceleration:
The next step in the kinetic ladder is acceleration. Acceleration
is the measurement of change in velocity per unit time, as shown in
Eq. 2a:
And since velocity is length divided by time:
Where:
a = Acceleration in meters per second squared. Therefore, acceleration is measured in "meters per second per second" or "meters per second squared".
2.2.5
Force:
Force is the next step in the kinetic ladder. When a force (such as
gravity) acts upon a mass, it produces acceleration as shown is Eq.
3a:
And since the fundamental units of acceleration are meters per
second squared, the dimensional equivalent in fundamental units is:
Where:
F = Force in meterkilograms per second squared. The physics unit of measure of force is the Newton. The common unit is the Pound (English). There is some confusion regarding the pound, since it is also used as a unit of weight. However, technically speaking "weight" is a measurement of the "force" of gravity. The Newton is defined as the force required to accelerate 1 kilogram at the rate of 1 meter per second squared. The Earth's gravitational field produces a force of 9.8 Newtons per kilogram.
2.2.6
Energy:
The next step on the kinetic ladder is energy, and is defined as
"force acting over a distance", as shown in Eq. 4a.
And since the dimensional equivalent of force is meterkilograms per
seconds squared (Eq. 3b), the dimensional units of energy are:
Where:
E = Energy in meters squaredkilograms per second squared. The physics unit of measure for energy is the Joule. Other common units of energy are FootPounds, or WattSeconds. The Joule is defined as one Newton of force, acting over a distance of one meter. The FootPound is self evident (the wattsecond will become clear shortly). Note: In thermodynamics, the variable Q is commonly used to denote energy, rather than E.
2.2.7
Power:
The final step on the kinetic ladder is power. Power is defined as
energy per unit of time, as shown in Eq. 5a:
And since the dimensional equivalent of energy is meters
squaredkilograms per seconds squared, the dimensional units of work
are:
Where:
W = Power in meters squaredkilograms per second cubed. The physics unit of measure for power is the Watt (named after James Watt, of steam engine fame), and is defined as 1 Joule per second. The common unit of measure for power is horsepower. One horsepower is defined as 550 FootPounds per second. And since one Watt is one Joule per second, it follows that a Joule can also be defined as one WattSecond, as shown in Eq. 5c.
2.3
The finished ladder:
Here is the ladder of kinetic units:
Notice how each step is built upon the previous step, by multiplying or dividing the previous step by one of our fundamental units. Now you see why we are justified in calling this the "ladder of kinetic units". While far from a complete list of kinetic units of measure, this list represents the most commonly used units. Furthermore, all other kinetic units of measure can be viewed of as side branches, derived from this main ladder.
2.4
Summary:
Any statement in physical mathematics can be decomposed into a
combination of four fundamental units of measure (2.2.3, 2.2.4,
2.2.5, 2.2.6, 2.2.7). Furthermore, basic kinetic units of
measurement form an ascending ladder (2.3). In part 3, we shall
apply dimensional analysis to electromagnetic units of measure.
End.
Dimensional Analysis  Part 2
