Chapter III

If our designs for private houses are to be correct, we must at the outset take note of the countries and climates in which they are built.

- Vitruvius

It seemed necessary to make this point to enable me to deal with the objection that people of this kind would never be found on the Earth but only in the mind of a moon-struck individual, one-third mentally deranged by fantasies of space travel and two-thirds lost to realities in a maze of mathematics.

- Hermann Oberth

The physical laws necessary to describe and predict the properties of artificial gravity were formulated by Isaac Newton more than three hundred years ago.  Nevertheless, it has been my observation that many people outside the aerospace profession, including many architects, do not understand centripetal acceleration, weightlessness, or orbit.  This comes as no great surprise: these are not matters of daily concern for most people.  Yet while terrestrial architecture tends to take gravity for granted, extraterrestrial architecture cannot.  Designing for an alien environment requires an investigation of its properties and a reevaluation of previously-held assumptions in the light of universal physical laws.

In discussing this research with college-educated non-engineers, three questions have repeatedly been raised:

This chapter summarizes the relevant concepts from physics and engineering.  It begins with a review of Newton's laws of motion and gravitation, and a qualitative description of weightlessness and orbit.  Then, it discusses the relationship between gravity, acceleration, and weight.  Finally, it develops mathematical formulas for acceleration, momentum, energy, and structural stress.  The derivations rely on the notation of vector calculus: for those who understand it, it provides a concise expression of changing spatial relationships; for those who don't, it's merely hieroglyphics.  It is my hope that the accompanying text will make the general line of reasoning accessible to the less mathematically inclined.

The fundamental laws can be found in any introductory physics text; I have relied on Beiser [1] and Sears and Zemansky [2].  For guidance in the principles of dynamics, I have relied on texts by Shames [3], Hibbeler [4], and Pollard [5].  In presenting mathematical formulas I have followed their style: italics indicate scalar quantities; boldface indicates vector (magnitude and direction) quantities; dots above indicate derivatives with respect to time; absolute-values indicate magnitudes.  The following symbols are used throughout this chapter:

 R_boldposition vector

 R_bold_dotvelocity vector

 R_bold_dot_dotacceleration vector

 |R_bold|position magnitude (radius)

 |R_bold_dot|velocity magnitude (speed)

 |R_bold_dot_dot|acceleration magnitude

Furthermore, the upper-case forms R_bold, R_bold_dot, R_bold_dot_dot, and so on, represent measurements relative to global, inertial coordinate systems, while the lower-case forms r_bold, r_bold_dot, r_bold_dot_dot, and so on, represent measurements relative to local, possibly accelerated coordinate systems.  With regard to rotations:

 Ω boldangular velocity of a rotating coordinate system

 ω boldangular velocity of a body

 λ boldangular velocity of a body relative to a rotating coordinate system

These angular velocities may or may not be equal, depending on the context.  The distinction will be important.