This section begins with a proof that a body moving with purely centripetal acceleration follows a circular trajectory if the magnitude of the acceleration is held constant. It then uses the results of this proof to derive the most important equations in this chapter: a set of expressions for the position, velocity, and acceleration of a body in a rotating system, which are major determinants in the design of rotating habitats.
Centripetal acceleration, by definition, is perpendicular to velocity. Then the dot product of the velocity and acceleration vectors is zero:
Integrating both sides of this equation and solving for gives an expression for the magnitude of the velocity vector:
where the term is a constant of integration; therefore is constant. The velocity vector may be written as:
where is the direction of the velocity vector at a given time, and , are unit vectors directed parallel to the , coordinate axes. Taking the derivative of both sides gives an expression for the acceleration vector:
It has already been shown that is constant. If is held constant, then - the rate of change of direction - must also be constant. Let be represented by . Then substitution and integration give the following expressions for: the rate of change of direction, the instantaneous direction at time , the acceleration, velocity, and position vectors, and the magnitude of the position vector:
It has already been shown that both and are constant. Therefore - the magnitude of the position vector - is constant, and the body moves on a circular trajectory with radius , tangential velocity , and rotation rate (radians per unit time). (Non-zero constants of integration and would simply translate the center of the circle at constant velocity. We are free to choose a coordinate system that translates with the circle, such that these terms vanish.)
The expression represents the direction angle of the velocity vector at time , where the term is an arbitrary constant of integration. The expression for the position vector can be simplified by substituting for and setting equal to :
The angular velocity of the body can also be represented by a vector, with magnitude and direction perpendicular to the plane of rotation, according to the right-hand rule: form a "hitch-hiker's thumb" with the right hand; when the fingers curl in the direction of rotation, then the thumb indicates the direction of the angular velocity vector. This vector can be expressed as:
where is the rate of rotation, and is a unit vector directed parallel to the coordinate axis.
It is now possible to express the position, velocity, and acceleration in terms of the radius and rotation:
Equations 3.10 through 3.16 express in a nutshell the basic characteristics of rotational motion, and serve as the starting point for the design of artificial-gravity rotating environments. In particular, equation 3.16 shows that the apparent gravity level is directly proportional to the square of the rotation rate , and to the radius . Given the desired gravity level and rate of rotation, equation 3.16 virtually dictates the radius.
Equation 3.16 also contains the first evidence that this form of gravity is somewhat different from what we experience on Earth. The centripetal acceleration vector is aligned along the radius, and the magnitude of the acceleration is proportional to the radial distance. Therefore, a person standing in a rotating environment (aligned "vertically" with the centripetal acceleration vector) will experience a gravity differential from his head to his feet; his apparent net weight will be proportional to the distance from the center of rotation to his center of mass; he will gain weight when he lies down, and lose weight when he stands up. Relative to their "normal" Earth weights, short people will weigh proportionally more than tall people because their centers of mass will be farther from the center of rotation. Furthermore, since the human body is not a rigid body, the distribution of weight among its various moving parts must be considered. For example, the weights of the hands and feet will vary with their height while walking, stooping, lifting, and so on. All of these effects are minimized when the variation in radius is small with respect to the average radius, and may be negligible if the average radius is very large.
Of course, the Earth also rotates, and normal Earth gravity varies not just with distance but with distance squared. In theory, all of the effects described above are present in everyday Earthly life. However, the centripetal acceleration attributable to the Earth's rotation is incidental to the apparent gravity at its surface (on the order of 0.003 g at the equator, and less at higher latitudes), and variations in height are negligible compared to the average radius from the Earth's center (approximately 720 times the height of Mt. Everest). On the other hand, the apparent gravity in a rotating space station will be entirely due to centripetal acceleration (tidal accelerations may play a minor role), the radius will be smaller by a factor of thousands, and the angular velocity will be greater by a factor of thousands. It stands to reason that the effects of centripetal acceleration will be far greater in a rotating space station than they are on the Earth.