The gravitational force experienced by an object is called its weight. The force acts whether or not the object is in motion, but it is most conveniently measured when the object is at rest. Every scale to record weight is a force-measuring device. The units shown on the readout are usually mass units, not force units (a permissible substitution that I discuss below). Since this measurement of weight is a static measurement, it provides important information about a fundamental force of nature independent of the laws of motion.
If, armed with a scale and some curiosity, you set about learning all that you could about gravity, you would probably wish to seek answers to the following questions: (1) How does the gravitational force experienced by a given object depend on the location of the object? (2) Does the force on a particular object depend on its state of motion? (3) How does the force vary from one object to another? A little experimentation would reveal that the answer to the first question is, to a good approximation: Not at all. Your weight is the same upstairs as down, the same in one city as in another. Actually precise experiments reveal slight weight difference from one part of the Earth to another, and a gradual decrease of weight with increasing altitude. However, if you concluded that every object near the surface of the earth experiences a definite fixed gravitational force regardless of its location, you would be very close to the truth.
The dependence of the gravitational force on motion is less easy to measure in general, but experiments reveal that it is independent of motion. You could, for example, weigh yourself on a uniformly moving train or a uniformly moving elevator and verify that your weight is unchanged. In accelerated motion, an ordinary scale would give a different reading. However, this comes about only because the scale, when accelerated, is no longer a suitable device for measuring the gravitational force. Other methods could reveal that even in accelerated motion, the gravitational force is unchanged.
Galileo considered both of these questions and answered them both, the first by measurement, the second by guess. He argued that since an object weighs the same at different heights, it probably continues to feel the same constant force as it falls through these different heights. In this way he connected the observed constant acceleration of a falling object to the hypothesized constant force acting on it.
Much can be learned about the variation of gravitational force from one object to another through static weight measurements, although the full clarification of this subject requires a study of motion as well. Common experience tells us that the weight of an object is somehow related to the amount of matter in it. For a particular kind of material, weight is proportional to volume. One pint of water weighs one pound (lb), one quart weighs two lb, and one gallon weighs eight lb.1 On the other hand, a gallon of gasoline weighs 6 lb, and, a gallon of lead weighs 109 lb. There is no universal connection between weight and volume. One comes considerably closer to a simple relationship in trying to establish a connection between weight and the number of nucleons (protons plus neutrons) in a piece of material. There is almost a direct proportion, but not quite. The ratio of the weight of an object to the number of its constituent nucleons varies by as much as eight parts in a thousand from one substance to another (the mass-energy equivalence is the culprit). However, weight has been found to be directly proportional to mass (see Essay M8). This proportionality, apparently universal for all substances of all sizes, has been tested with increasing accuracy over the years: by Isaac Newton, in the seventeenth century, to one part in 103; by Loránd Eötvös in the early years of the twentieth century, to one part in 108; by Robert Dicke and collaborators in the 1960s, to one part in 1011; and by scientists following the Moon’s orbit with exquisite precision in the early years of the twenty-first century, to a few parts in 1013.2
Because of the observed proportionality between weight and mass (inertial mass), it is easy to confuse these concepts. Inertial mass, defined as the constant of proportionality between force and acceleration in Newton’s second law, is a measure of the resistance to a change in motion of an object subjected to a force, any force. Weight, on the other hand, is specifically a measure of the intensity of response of an object to a gravitational force. Weight is, in fact, the gravitational force experienced by an object, regardless of its motion. There is no obvious reason at all for a connection between these two concepts. Their direct proportionality, discovered by Newton, remained a mysterious coincidence for over two centuries—finally resolved by Einstein’s general theory of relativity.
The fact that objects near the earth on which the frictional force is negligible all fall with the same constant acceleration is so familiar that its significance is easily overlooked. It ought to be regarded as truly remarkable. Consider a number of different weights. On each the pull of gravity is different. Yet all fall with precisely the same acceleration. How does it come about that a body that experiences twice the gravitational force of another also has exactly twice the inertial mass so that both move in exactly the same way? It is not my purpose here to explain this coincidence, only to re-emphasize that there is no obvious reason for mass and the magnitude of a gravitational force to be proportional. No other fundamental forces in nature are simply related to mass. Proton and positron, for example, experience the same electric force despite a large difference in their mass. Proton and neutron, with nearly the same mass, feel very different electric and magnetic forces.
The puzzle of the proportionality of weight to mass came into prominence in science in the latter part of the nineteenth century. In particular, a question arose whether the ratio of weight to mass at a particular spot on the earth is truly constant or varies slightly from one substance to another. In order to facilitate discussion of this question and to highlight the puzzle, a new concept, gravitational mass, was introduced. The meaning of gravitational mass can best be made clear by considering a hypothetical experiment. Imagine yourself in a laboratory equipped to measure force and velocity and acceleration, with a number of different objects at your disposal, including a standard kilogram. Since the experiment is hypothetical, imagine too that you have available a perfectly flat frictionless horizontal surface upon which to slide the objects. A known horizontal force (not a gravitational force) is applied to each object in turn, and the acceleration measured. Through Newton’s second law, the force and acceleration provide a measurement of the inertial mass, mI (the subscript I is introduced to emphasize that it is inertial mass that is being measured):
mI = F/a.
Having defined and measured the inertial mass of each object by means of experiments independent of gravity, you turn attention to the nature of the gravitational force acting on each of your test objects. Here a fundamental postulate enters. Its truth will be tested by further experiments. The postulate is this: The magnitude of gravitational force experienced by a particular object depends on only one single property of that object, a scalar quantity whose numerical magnitude is directly proportional to the strength of the object’s gravitational interaction. This is obviously a powerful postulate about the simplicity of nature. It says that a single number is enough to reveal how much gravitational force a body will experience, regardless of its shape or color or chemical composition, or whether it contains liquid, solid, or gas. To this numerical quantity the name gravitational mass is given. It is to be regarded as an intrinsic property of a body, completely analogous to electric charge. Whereas charge measures the electric interaction strength of a body, gravitational mass measures the gravitational interaction strength. Indeed gravitational charge might be a better name for the quantity.
According to the postulate, the gravitational force experienced by any one of the test objects should be proportional to its gravitational mass, mG. The mathematical expression of the postulate is
Fgrav = gmG,
where g is a constant of proportionality.3 The new concept, gravitational mass, needs a unit of measurement and a standard for this unit. As a matter of convenience, you may decide to measure gravitational mass in the same unit as inertial mass, the kilogram, and to adopt as its standard the same one-kilogram object. Having done this, you proceed to weigh each of your test objects, that is, to determine by a static measurement the gravitational force on each. The weight of the standard kilogram determines the constant g through the equation
mG = Fgrav/g.
And, of course, g is the acceleration of gravity for all objects at that place. This implies, using Newton’s second law, that mI = Fgrav/g, whence
mI = mG.
1 The pound is a unit of mass (equal to 0.454 kg). When we say that an object “weighs” 2 lb, we mean that its mass has been so determined by measuring the gravitational force acting on it.
2 Newton took note of the fact that the period of a pendulum of a given length does not depend on the mass being swung. Eötvös and Dicke sought (and failed to find) any difference in the acceleration of different substances on Earth toward the Sun. Recent studies use what is called lunar laser ranging to compare the accelerations of Moon and Earth toward the Sun with what they would be if the accelerations depended on mass.
3 The quantity g may depend on where the experiment is carried out. It is a constant only at a given place. The more essential fact about g is that it is independent of all properties of the object experiencing the force. The gravitational mass mG depends on the object; g does not.