3.3    Gravity, Acceleration, and Weight

The relationship between gravity, acceleration, and weight can be examined by considering the following four situations:

Figure 3.4.1
Figure 3.4.2
Figure 3.4.3
Figure 3.4.4

Figure 3.4:  Gravity, acceleration, and weight.

1)  Consider a person near but somewhat above the surface of the Earth.  Since his size (approximately 2 meters) is insignificant compared to his distance from the Earth's center (approximately 6,380,000 meters), every particle of his body is essentially equidistant from the Earth's center, and undergoes the same acceleration under the influence of gravity.  Gravity causes him to accelerate uniformly, without producing any internal stresses that could provide the sensation of weight.

2)  Now consider this person standing on the surface of the Earth.  (Assume that he survives the impact and rapid deceleration in good shape, and reaches a state of equilibrium.)  Gravity continues to pull down on every particle of his body, but now his continued downward motion is thwarted by the bulk of the Earth, which pushes up with a force equal and opposite to the total pull of gravity.  Unlike gravity, this upward push is not applied directly to every part of the body, but only to the feet.  From there, it propagates through the body, with each part supporting every part above.  The body undergoes non-uniform compression, which is maximum at the bottom of the feet, diminishes to zero at the top of the head, and affects every organ in between.

3)  Now suppose that, for unexplained reasons, gravity suddenly disappears, but the ground continues to push up on this person with the same force as before.  By the second law of motion, the person and his patch of ground must accelerate upward at a uniform rate.  The accelerating force is applied directly only to the feet, and from there it propagates through the body, with each part accelerating every part above.  Because of the equality of gravitational and inertial mass, the distribution of stresses within the body is exactly as before:  maximum compression at the bottom of the feet, diminishing to zero at the top of the head, and affecting every organ in between.

4)  Finally, suppose that an equal and opposite (but non-gravitational) force is applied to the top of the head.  The acceleration stops.  The person is still under compression, but this vice-like compression is unlike what he experienced earlier: it is essentially uniform from head to foot, and imposed exclusively on the skeletal system, leaving the other organs largely unaffected.

The first two situations might seem to refute the second law of motion: in the first case, the person accelerates, though he feels no force; in the second case, the person feels a force, but fails to accelerate.  However, there is nothing in the second law that requires a body to perceive the forces acting upon it.

In the second situation, it is useful to think of the acceleration not simply as zero, but as the sum of two non-zero components: a downward acceleration due to gravity, and an equal and opposite upward acceleration due to the bulk of the Earth.

An analysis of these four situations shows that the perception of weight is quite independent of gravity.  In fact, perceived or apparent weight can be thought of as inertial resistance to every accelerating force except gravity.  We perceive weight not because gravity pulls us down, but because the ground pushes us up.  To compute the apparent weight of a body, begin by subtracting its gravitational acceleration from its total acceleration.  The apparent weight is equal and opposite to the mass times the non-gravitational acceleration:

[math]

(3.6)

where:

 w'  is the apparent weight of the body.

 m  is the mass of the body.

 R_dot_dot  is the total (net) acceleration of the body due to all forces (gravitational and other) acting upon it.

 R_dot_dot_g  is the gravitational component of R_dot_dot, computed according to equation 3.3.

Equation 3.6 states that the apparent weight of a body is completely determined by its mass and non-gravitational acceleration - there are no other factors [7].  Clark and Hardy [8] discuss the inadequacies of magnetic shoes and other restraint mechanisms for producing apparent weight:  "To produce orienting forces throughout the tissue, acceleration must be used."

The term "artificial gravity" may be a bit of a misnomer.  In a chapter entitled "The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity" [9], Einstein describes a thought experiment:  In a large region of empty space, far removed from any appreciable mass, a large chest containing an observer is accelerated upward by a "being".  Every experiment the observer can perform within the confines of the chest indicates that the chest is suspended motionless in a gravitational field.  Einstein goes on to state that "a gravitational field exists for the man in the chest, despite the fact that there was no such field for the coordinate system first chosen."

Much of the remainder of this chapter is devoted to an analysis of acceleration, which is fundamental for understanding artificial gravity and apparent weight.