Weightlessness - Weightlessness in Newtonian Mechanics

Weightlessness in Newtonian Mechanics

In Newtonian mechanics the term "weight" is given two distinct interpretations by engineers.

Weight₁: Under this intepretation, the "weight" of a body is the gravitational force exerted on the body and this is the notion of weight that prevails in engineering. Near the surface of the earth, a body whose mass is 1 kg has a weight of approximately 10 N, independent of its state of motion, free fall, or not. Weightlessness in this sense can be achieved by removing the body far away from the source of gravity. It can also be attained by placing the body at a neutral point between two gravitating masses.

Weight₂: Weight can also be interpreted as that quantity which is measured when one uses scales. What is being measured there is the force exerted by the body on the scales. In a standard weighing operation, the body being weighed is in a state of equilibrium as a result of a force exerted on it by the weighing machine cancelling the gravitational field. By Newton's 3rd law, there is an equal and opposite force exerted by the body on the machine. This force is called weight₂. The force is not gravitational. Typically, it is a contact force and not uniform across the mass of the body. If the body is placed on the scales in a lift (an elevator) in free fall in pure uniform gravity, the scale would read zero, and the body said to be weightless i.e. its weight₂ =0. This is the dominant notion of weightlessness in engineering discourse. It describes the condition in which the body is stress free and undeformed. This is the weightlessness in free fall in a uniform gravitational field. (The situation is more complicated when the gravitational field is not uniform, or, when a body is subject to multiple forces which may, for instance, cancel each other and produce a state of stress albeit weight₂ being zero. See below.)

To sum up, we have two notions of weight of which weight₁ is dominant. Yet 'weightlessness' is typically exemplified not by absence of weight₁ but by the absence of stress associated with weight₂. This is the intended sense of weightlessness in what follows below.

A body is stress free, exerts zero weight₂, when the only force acting on it is weight₁ as when in free fall in a uniform gravitational field. Without subscripts, one ends up with the odd sounding conclusion that a body is weightless when the only force acting on it is its weight.

The apocryphal apple that fell on Newton's head can be used to illustrate the issues involved. An apple weighs approximately 1 newton. This is the weight₁ of the apple and is considered to be a constant even while it is falling. During that fall, its weight₂ however is zero: ignoring air resistance, the apple is stress free. When it hits Newton, the sensation felt by Newton would depend upon the height from which the apple falls and weight₂ of the apple at the moment of impact may be many times greater than 1 N. It was great enough—in the story—to make the great man invent the theory of gravity. It is this weight₂ which distorts the apple. On its way down, the apple in its free fall does not suffer any distortion as the gravitational field is uniform.