Mass

MASS (in ordinary understanding), the quantity of substance contained in a given body; the exact definition follows from the basic laws of mechanics. According to Newton's second law, "the change of motion is proportional to the acting force and has the same direction with it." By "change of motion" here is meant the change in the velocity of motion. Thus, if we have some body, for example, a ball, and we give it pushes (i.e., we act on it with instantaneous forces) of different intensities, the ball will begin to move with greater velocity (i.e., will acquire greater acceleration) the greater the intensity of the push. Let us now imagine that we have two bodies, for example, two balls, of the same substance but of different diameters, or of the same diameter but of different substances, for example, one wooden and the other leaden, etc. If we give both pushes of equal force, experience teaches that of the two balls of different diameters, the smaller will acquire greater velocity; of two balls of the same diameter but of different substances, for example, the wooden ball will acquire greater velocity than the leaden, etc. Generalizing, we can say that the acceleration imparted by a force is determined not only by the magnitude of the force but also by some property of the body to which this force is applied. This property can be called the inertia of the body; from a mechanical point of view, it is precisely this that serves as the measure of the mass of the body. Returning to the example of balls of the same substance but of different diameters, we again, on the basis of experience, can establish that under the action of an equal force, the ball will acquire acceleration so many times greater as its volume is smaller than the volume of the other ball. Thus, in the final analysis, we can assert that the acceleration w must be directly proportional to the force F and inversely proportional to the mass M; i.e., w = F/M, from which follows the definition of mass M = F/w. It is necessary to note that with this form of definition of mass, we are already bound in the choice of units: of the three quantities entering into the written equality, only for two can we choose units arbitrarily; the unit of the third quantity will thereby already be fixed. For example, having arbitrarily chosen the unit of mass and acceleration, we fix the unit of force, since from the equality 1 = F/M follows that F = 1, etc. In the absolute CGS system of units (see Absolute system of measures), the units of length, time, and mass have been chosen as the basic ones. The absolute unit of mass is the gram. Although the strict definition of mass is based on the comparison of accelerations, in practice when measuring the mass of bodies, their weights are compared, i.e., the forces with which bodies are attracted to the earth. Such a replacement is possible thanks to one peculiarity of the force of gravity, which places it in an exceptional position. Returning once more to those mental experiments by means of which we established the unit of mass, we first of all notice that there is a parallelism between weight and mass: heavier bodies also possess greater mass. This parallelism is strictly quantitative; its consequence is the well-known fact that all bodies in a vacuum fall with the same velocity, i.e., that the acceleration of gravity g is constant for all bodies: F = Mg or g = F/M, where F is the weight of the body and g is the acceleration of gravity, the same for all bodies. Since g is a constant quantity, the written formula contains the statement that the weight of a body is strictly proportional to its mass, a statement that allows the comparison of mass to be reduced to the comparison of weights. This remarkable property of the force of gravity, since Galileo's discovery of the independence of the acceleration g from mass, remained completely mysterious and incomprehensible for centuries. Only in the general theory of relativity did this fact acquire the significance of a cornerstone of the entire theory. The weight of a body is not an absolutely constant quantity, but depends on the tension of the force of gravity and therefore depends on the geographical position of the place (at the equator it is somewhat less than at the poles). On the contrary, mass is a property inalienably inherent in the body itself and does not depend on the physical conditions in which the body is found. The so-called law of conservation of matter is in essence the law of conservation of mass. Until comparatively recent times, mass was generally attributed the value of an absolutely constant constant. The theory of relativity introduced here very essential corrections. First, it turned out that the mass of a body depends on its velocity. This dependence is expressed by the formula m = m₀/√(1-v²/c²), where m₀ is the mass of the body at rest, v is the velocity of the body, and c is the velocity of light. Since the latter is expressed by the huge number 3×10¹⁰ cm/sec, and the maximum velocities with which one has to deal practically on earth are of the order of magnitude 10²-10³ cm/sec, the denominator of the fraction is very close to unity, from which it follows that at ordinary terrestrial velocities (including the velocities of airplanes and artillery projectiles) mass practically does not depend on velocity. But for α- and β-particles of radium, ejected with velocities up to 1/3 the velocity of light, mass already depends substantially on velocity. Experiments performed with β-particles of radium and with fast electrons in discharge tubes showed that the formula for the dependence of mass on velocity given by the theory of relativity is confirmed with great accuracy. The second correction is due to the fact that, as the theory of relativity shows, energy must also be attributed a mass equal to m' = E/c², where E is energy and c is the velocity of light. Thus, theoretically speaking, a heated body should possess somewhat greater mass than a cooled one, since in the first case the energy reserve of the body is greater. However, if we take into account that in the denominator of the formula stands the square of the velocity of light, i.e., a huge quantity (9×10²⁰), it is clear that with changes in energy of ordinary scale, this correction can be neglected. However, already in radioactive phenomena it reaches a quite perceptible magnitude. The release of enormous quantities of energy in the formation of atomic nuclei at the present time explains the fact that the atomic masses of our elements or their isotopes (see Atom) are not exact multiples of the mass of the proton (i.e., the nucleus of the hydrogen atom). For example, the mass of the helium atom, the nucleus of which is built from four protons, is 4.000, whereas the mass of the proton in the free state is 1.008. The difference between 1.008×4=4.032 and the mass of the helium nucleus (4.032-4.000=0.032) represents the so-called "mass defect," which is the result of the fact that in the combination of four protons and two electrons into the helium nucleus, an enormous quantity of energy was released, equal to 0.032×9×10²⁰ ergs, or 6.3×10⁹ b. cal. The grandeur of this number becomes especially clear if we recall that the ordinary heats of formation of chemical compounds have the order of magnitude of 100 b. cal. The enormous magnitude of the "formation energy" of the helium nucleus, approximately 3 times greater than the energy of the fastest α-particle, explains its exceptional stability.