Reference: Evolution of Physics
This paper presents Chapter I, section 5 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).
The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding. Feedback on these comments is appreciated.
The heading below is linked to the original materials.
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One Clue Remains
When first studying mechanics one has the impression that everything in this branch of science is simple, fundamental and settled for all time. One would hardly suspect the existence of an important clue which no one noticed for three hundred years. The neglected clue is connected with one of the fundamental concepts of mechanics—that of mass.
In mechanics, the concept of mass has not been explored fully.
Again we return to the simple idealized experiment of the cart on a perfectly smooth road. If the cart is initially at rest and then given a push, it afterwards moves uniformly with a certain velocity. Suppose that the action of the force can be repeated as many times as desired, the mechanism of pushing acting in the same way and exerting the same force on the same cart. However many times the experiment is repeated, the final velocity is always the same. But what happens if the experiment is changed, if previously the cart was empty and now it is loaded? The loaded cart will have a smaller final velocity than the empty one. The conclusion is: if the same force acts on two different bodies, both initially at rest, the resulting velocities will not be the same. We say that the velocity depends on the mass of the body, being smaller if the mass is greater.
As mass increases, the same push results in lesser velocity.
We know, therefore, at least in theory, how to determine the mass of a body or, more exactly, how many times greater one mass is than another. We have identical forces acting on two resting masses. Finding that the velocity of the first mass is three times greater than that of the second, we conclude that the first mass is three times smaller than the second. This is certainly not a very practical way of determining the ratio of two masses. We can, nevertheless, well imagine having done it in this, or in some similar way, based upon the application of the law of inertia.
By pushing and measuring velocities we may determine the mass. This is called inertial mass.
How do we really determine mass in practice? Not, of course, in the way just described. Everyone knows the correct answer. We do it by weighing on a scale.
By weighing on a scale also we may determine the mass. This is called gravitational mass
Let us discuss in more detail the two different ways of determining mass.
The first experiment had nothing whatever to do with gravity, the attraction of the earth. The cart moves along a perfectly smooth and horizontal plane after the push. Gravitational force, which causes the cart to stay on the plane, does not change, and plays no role in the determination of the mass. It is quite different with weighing. We could never use a scale if the earth did not attract bodies, if gravity did not exist. The difference between the two determinations of mass is that the first has nothing to do with the force of gravity while the second is based essentially on its existence.
The force of gravity is essential to gravitational mass, but it has nothing to do with inertial mass.
We ask: if we determine the ratio of two masses in both ways described above, do we obtain the same result? The answer given by experiment is quite clear. The results are exactly the same! This conclusion could not have been foreseen, and is based on observation, not reason. Let us, for the sake of simplicity, call the mass determined in the first way the inertial mass and that determined in the second way the gravitational mass. In our world it happens that they are equal, but we can well imagine that this should not have been the case at all. Another question arises immediately: is this identity of the two kinds of mass purely accidental, or does it have a deeper significance? The answer, from the point of view of classical physics, is: the identity of the two masses is accidental and no deeper significance should be attached to it. The answer of modern physics is just the opposite: the identity of the two masses is fundamental and forms a new and essential clue leading to a more profound understanding. This was, in fact, one of the most important clues from which the so-called general theory of relativity was developed.
It so happens that the two masses are exactly the same. The fundamental significance of this equality led to the theory of relativity.
A mystery story seems inferior if it explains strange events as accidents. It is certainly more satisfying to have the story follow a rational pattern. In exactly the same way a theory which offers an explanation for the identity of gravitational and inertial mass is superior to one which interprets their identity as accidental, provided, of course, that the two theories are equally consistent with observed facts.
Rational explanation is vital to any theory.
Since this identity of inertial and gravitational mass was fundamental for the formulation of the theory of relativity, we are justified in examining it a little more closely here. What experiments prove convincingly that the two masses are the same? The answer lies in Galileo’s old experiment in which he dropped different masses from a tower. He noticed that the time required for the fall was always the same, that the motion of a falling body does not depend on the mass. To link this simple but highly important experimental result with the identity of the two masses needs some rather intricate reasoning.
Different masses take the same time to fall from the same height. Therefore the motion of a falling body is independent of its mass.
A body at rest gives way before the action of an external force, moving and attaining a certain velocity. It yields more or less easily, according to its inertial mass, resisting the motion more strongly if the mass is large than if it is small. We may say, without pretending to be rigorous: the readiness with which a body responds to the call of an external force depends on its inertial mass. If it were true that the earth attracts all bodies with the same force, that of greatest inertial mass would move more slowly in falling than any other. But this is not the case: all bodies fall in the same way. This means that the force by which the earth attracts different masses must be different. Now the earth attracts a stone with the force of gravity and knows nothing about its inertial mass. The “calling” force of the earth depends on the gravitational mass. The “answering” motion of the stone depends on the inertial mass. Since the “answering ” motion is always the same all bodies dropped from the same height fall in the same way it must be deduced that gravitational mass and inertial mass are equal.
Different inertial masses are responding in the same way. This response is to the force of gravity that is acting on different gravitational masses. Since the “call” and the “response” are coordinated exactly the same way for different masses, the gravitational and inertial mass must be the same.
More pedantically a physicist formulates the same conclusion: the acceleration of a falling body increases in proportion to its gravitational mass and decreases in proportion to its inertial mass. Since all falling bodies have the same constant acceleration, the two masses must be equal.
In our great mystery story there are no problems wholly solved and settled for all time. After three hundred years we had to return to the initial problem of motion, to revise the procedure of investigation, to find clues which had been overlooked, thereby reaching a different picture of the surrounding universe.
It was this two different ways of looking at mass that provided a new clue.
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FINAL COMMENTS
The inertial mass is measured using the force of the push; whereas, the gravitational mass is measured using the force of gravity. In both cases mass is measured through the use of force. The only difference is that a push supplies force for a short time; whereas, the force of gravity is continuous in nature. As far as the duration of the experiment is concerned, however, the effect created on the body by the two forces is the same.
Now we have another clue: Inertia ia an internal force that is also continuous like the force of gravity. The inertia not only keeps the motion uniform but it also brings that motion in line with the mass.
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