Dimensional Analysis - Part 2. Of oranges, nukes, and virgins by Steven J. Smith

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.

1.	The ability to determine the validity of any statement of physical mathematics.
2.	The ability to see at a glance, how any statement in physical mathematics relates to other collateral statements, and how the statement fits into the larger paradigm we call physics.

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:

1.	Length. (feet, meters, etc.)
2.	Time. (seconds, hours, etc.)
3.	Mass. (grams, kilograms, etc.)
4.	Electric charge. (coulombs)

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 = MC²

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.

[Eq. 1]

(dimensional equivalent of velocity in fundamental units)

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:

[Eq. 2a]

And since velocity is length divided by time:

[Eq. 2b]

(dimensional equivalent of acceleration in fundamental units)

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:

[Eq. 3a]

And since the fundamental units of acceleration are meters per second squared, the dimensional equivalent in fundamental units is:

[Eq. 3b]

(dimensional equivalent of force in fundamental units)

Where:
F = Force in meter-kilograms 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.

[Eq. 4a]

And since the dimensional equivalent of force is meter-kilograms per seconds squared (Eq. 3b), the dimensional units of energy are:

[Eq. 4b]

(dimensional equivalent of energy in fundamental units)

Where:
E = Energy in meters squared-kilograms per second squared.

The physics unit of measure for energy is the Joule. Other common units of energy are Foot-Pounds, or Watt-Seconds. The Joule is defined as one Newton of force, acting over a distance of one meter. The Foot-Pound is self evident (the watt-second 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:

[Eq. 5a]

And since the dimensional equivalent of energy is meters squared-kilograms per seconds squared, the dimensional units of work are:

[Eq. 5b]

(dimensional equivalent of power in fundamental units)

Where:
W = Power in meters squared-kilograms 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 Foot-Pounds per second.

And since one Watt is one Joule per second, it follows that a Joule can also be defined as one Watt-Second, as shown in Eq. 5c.

[Eq. 5c]

2.3

The finished ladder:

Here is the ladder of kinetic units:

1.	Velocity is length per unit time.
2.	Acceleration is change in velocity per unit time.
3.	Force is acceleration multiplied by mass.
4.	Energy is force multiplied by length.
5.	Power is energy per unit time.

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