Kitabı oku: «The Energy System of Matter: A Deduction from Terrestrial Energy Phenomena», sayfa 5
20. Transformations—Upward Movement of a Mass against Gravity
When the significance of energy inception and the characteristic properties of the various agencies have been grasped, it becomes much easier to deal with certain other aspects of energy processes. To illustrate these aspects it is, therefore, now proposed to discuss a few simple secondary operations embodied in experimental apparatus. A few examples of the operations of transformation and transmission of energy will be considered. The object in view is to show the general nature of these processes, and more especially the limits imposed upon them by the various factors or properties of the material machines in which they are of necessity embodied. The reader is asked to bear in mind also the observations already made (§ 13) with respect to experimental apparatus generally.
The first operation for discussion is that of the upward movement of a mass of material against the gravitative attraction of the earth. This movement involves one of the most simple and at the same time one of the most important of secondary energy processes. As a concrete illustration, consider the case of a body projected vertically upwards with great velocity from the surface of the earth. The phenomena of its motion will be somewhat as follows:—As the body recedes from the earth's surface in its upward flight, its velocity suffers a continuous decrease, and an altitude is finally attained where this velocity becomes zero. The projectile, at this point, is instantaneously at rest. Its motion then changes; it commences to fall, and to proceed once more towards the starting-point with continuously increasing velocity. Neglecting the effect of the air (§ 29) and the rotational movement of the earth, it may be assumed that the retardation of the projectile in its upward flight is numerically equal to its acceleration in its downward flight, and that it finally returns to the starting-point with velocity numerically equal to the initial velocity of projection. The process then obviously involves a complete transformation and return of energy. At the earth's surface, where its flight commences and terminates, the body is possessed of energy of motion to a very high degree. At the highest point of flight, this form of energy has entirely vanished; the body is at rest. Its energy properties are then represented by its position of displacement from the earth's surface; its energy of motion in disappearing has assumed this form of energy of position, energy of separation, or potential energy. The moving body has thus been the mechanism of an energy transformation. At each stage of its upward progress, a definite modicum of its original energy of motion is converted into energy of position. Between the extreme points of its flight, the energy of the body is compounded of these two forms, one of which is increasing at the expense of the other. When the summit of flight is reached the conversion into energy of position is complete. In the downward motion, the action is completely reversed, and when the body reaches the starting-point its energy of position has again been completely transformed into energy of motion. It might be well to draw attention here to the fact, often overlooked, that this energy of position gained by the rising mass is, in reality, a form of energy, separate and distinct, brought into existence by the transformation and disappearance of the energy of the moving mass. Energy of position is as truly a form of energy as heat or kinetic energy.
The transformation here depicted is clearly a simple process, yet we know absolutely nothing of its ultimate nature, of the why or wherefore of the operation. Our knowledge is confined to the circumstances and conditions under which it takes place. Let us now, therefore, deal with these conditions. The transformation is clearly carried out in virtue of the movement of the body in the lines or field of an incepting influence. This influence is that of gravitation, which links the body continually to the earth. Now the function of gravitation in this process, as in others already described, is that of a completely passive incepting agent. The active energy which suffers change in the process is clearly the original work energy (§ 31) communicated to the projected body. The whole process is, in fact, a purely mechanical operation, and as in the case of other processes involving mechanical energy, it is limited by the mass value of the moving material. It is clear that the greater the amount of energy communicated to the projectile at the starting-point, the greater will be the altitude it will attain in its flight. The amount of energy, however, which can thus be communicated is dependent on the maximum force which can be applied to the projectile. But the maximum force which can be applied to any body depends entirely on the resistance offered by that body, and in this case the resisting force is the gravitative attraction of the earth on the projectile, which attraction is again a direct function of its mass. The greater the mass, the greater the gravitative force, and the greater the possibility of transformation. The ultimate limit of the process would be reached if the projected mass were so great as to equal half the mass of the earth. In such circumstances, the earth being assumed to be divided into two equal masses, the maximum limiting value of the gravitative attraction would clearly be attained. Any increase of the one mass over the other would again lead, however, to a diminution in the attractive force and a corresponding decrease in the energy limit for transformation. The precise manner in which the operations of mechanical energy are limited by the mass will now be clear. The principle is quite general, and applicable to all moving bodies. Mass is ever a direct measure of energy capacity. A graphical method of representing energy transformations of this kind, by a system of co-ordinates, would enable the reader to appreciate more fully the quantitative relations of the forms of energy involved and also their various limits.
21. The Simple Pendulum
The remaining operations of transformation for discussion are embodied in the following simple apparatus. A spherical metallic mass M (Fig. 2) is supported by a rod P which is rigidly connected to a horizontal spindle HS as shown.
Fig. 2
The spindle is supported and free to revolve in the bearings B1 and B2 which form part of the supporting framework V resting on the ground; the bearing surfaces at B1 and B2 are lubricated, and the mass M is free to perform, in a vertical plane, complete revolutions about the axis through the centre of the spindle. In carrying out this motion its path will be circular, as shown at DCFE; the whole arrangement is merely an adaptation of the simple pendulum. As constituted, the apparatus may form the seat of certain energy operations. Some of these will only take place with the application of energy of motion to the pendulum from an external source, thereby causing it to vibrate or to rotate: others, again, might be said to be inherent to the apparatus, since they arise naturally from its construction and configuration. We shall deal with the latter first.
22. Statical Energy Conditions
The pendulum with its spindle has a definite mass value, and, assuming it to be at rest in the bearings B1 and B2, it is acted upon by gravitation, or in other words, it is under the influence or within the field of the gravitative attraction of the earth's mass upon it. The effect of this field is directly proportional to the mass of the pendulum and spindle, and to its action is due that bearing pressure which is transmitted through the lubricant to the bearing surfaces and thence to the supporting arms N1 and N2 of the framework. Bearings and columns alike are thus subjected to a downward thrust or pressure. Being of elastic material, they will be more or less distorted. This distortion will proceed until the downward forces are balanced by the upward or reactive forces called into play in virtue of the cohesive properties of the strained material. Corresponding to a slight downward movement of the pendulum and spindle in thus straining or compressing them, the supporting columns will be decreased in length. This downward movement is the external evidence of certain energy operations. In virtue of their elevation above the earth's surface, the pendulum and spindle possess, to a certain degree, energy of position, and any free downward movement would lead to the transformation of this energy into energy of motion (§ 20). But the downward motion of pendulum and spindle is not free. It is made against the resistance of the material of the supporting columns, and the energy of position, instead of assuming the form of energy of motion, is simply worked down or transformed against the opposing cohesive forces of the supporting materials. This energy, therefore, now resides in these materials in the form of energy of strain or distortion. In general nature, this strain energy is akin to energy of position (§ 20). Certain portions of the material of the columns have been forced into new positions against the internal forces of cohesion which are ever tending to preserve the original configuration of the columns. This movement of material in the field of the cohesive influence involves the transformation of energy (§ 4), and the external evidence of the energy process is simply the strained or distorted condition of the material. If the latter be released, and allowed to resume its natural form once more, this stored energy of strain would be entirely given up. In reality, the material can be said to play the part of a machine or mechanism for the energy process of storage and restoration. No energy process, in fact, ever takes place unless associated with matter in some form. The supporting arms, in this case, form the material factor or agency in the energy operation. All such energy machines, also, are limited in the extent of their operation, by the qualities of the material factors. In this particular case, the energy compass of the machine is restricted by certain physical properties of the material, by the maximum value of these cohesive or elastic forces called into play in distortion. These forces are themselves the evidence of energy, of that energy by virtue of which the material possesses and maintains its coherent form. In this case this energy is also the factor controlling the transformation, and appears as a separate and distinct incepting agency. If the process is to be a reversible one, so that the energy originally stored in the material as strain energy or energy of distortion may be completely returned, the material must not be stressed beyond a certain point. Only a limited amount of work can be applied to it, only a limited amount of energy can be stored in it. Too much energy applied—too great a weight on the supporting columns—gives rise to permanent distortion or crushing, and an entirely new order of phenomena. This energy limit for reversibility is then imposed by the cohesive properties of the material or by its elastic limits. Up to this point energy stored in the material may be returned—the process is reversible in nature—but above this elastic limit any energy applied must operate in an entirely different manner.
A little consideration will show also, that the state of distortion, or energy strain, is not confined to the material of the supporting columns alone. Action and reaction are equal. The same stresses are applied to the spindle through the medium of bearings and lubricant. In fact, every material substance of which the pendulum machine is built up is thus, more or less, strained against internal forces; all possess, more or less, cohesion or strain energy. It will be evident, also, that this condition is not peculiar to this or any other form of apparatus. It is the energy state or condition of every structure, either natural or artificial, which is built up of ordinary material, and which, on the earth's surface, is subjected to the influence of the gravitation field. This cohesion or strain energy is one of the forms in which energy is most widely distributed throughout material.
In reviewing the statical condition of the above apparatus, the pendulum itself has been assumed to be hanging vertically at rest under the influence of gravitation. If energy be now applied to the system from some external source so that the pendulum is caused to vibrate, or to rotate about the axis of suspension, a new set of energy processes make their appearance. The movement of the pendulum mass, in its circular path around the central axis, is productive of certain energy reactions, as follows:—
a. A transformation of energy of motion into energy of position and vice versa.
b. A frictional transformation at the bearing surfaces.
These processes will each be in continuous operation so long as the motion of the pendulum is maintained. Their general nature is quite independent of the extent of that motion, whether it be merely vibratory through a small arc, or completely rotatory about the central axis. In the articles which immediately follow, the processes will be treated separately.
23. Transformations of the Moving Pendulum—
a. Energy of Motion to Energy of Position and Vice Versa
In this simple transformation the motion of the pendulum about the axis of suspension may be either vibratory or circular, according to the amount of energy externally applied. In each case, every periodic movement of the apparatus illustrates the whole energy operation. The general conditions of the process are almost identical with those in the case of the upward movement of a mass against gravity (§ 20). Gravitation is the incepting energy influence of the operation. If the pendulum simply vibrates through a small arc, then, at the highest points of its flight, it is instantaneously at rest. Its energy of motion is here, therefore, zero; its energy of position is a maximum. At the lowest point of its flight, the conditions are exactly reversed. Here its energy of motion is a maximum, while its energy of position passes through a minimum value. The same general conditions hold when the pendulum performs complete revolutions about the central axis. If the energy of motion applied is just sufficient to raise it to the highest point E (Fig. 2), the mass will there again be instantaneously at rest with maximum energy of position. As the mass falls downwards in completing the circular movement, its energy of position once more assumes the kinetic form, and reaches its maximum value at C (Fig. 2), the lowest position. The moving pendulum mass, so far as its energy properties are concerned, behaves in precisely the same manner as a body vertically projected in the field of the gravitative attraction (§ 20). This simple energy operation of the pendulum is perhaps one of the most familiar of energy processes. By its means, however, it is possible to illustrate certain general features of energy reactions of great importance to the author's scheme.
The energy processes of the pendulum system are carried out through the medium of the material pendulum machine, and are limited, both in nature and degree, by the properties of that machine. As the pendulum vibrates, the transformation of energy of motion to energy of position or vice versa is an example of a reversible energy operation. The energy active in this operation continually alternates between two forms of energy: transformation is continually followed by a corresponding return. Neglecting in the meantime all frictional and other effects, we will assume complete reversibility, or that the energy of motion of the pendulum, after passing completely into the form of energy of position at the highest point, is again completely returned, in its original form, in the descent. Now, for any given pendulum, the amount of energy which can thus operate in the system depends on two factors, namely, the mass of the pendulum and the vertical height through which it rises in vibration. If the mass is fixed, then the maximum amount of energy will be operating in the reversible cycle when the pendulum is performing complete revolutions round its axis of suspension. The maximum height through which the pendulum can rise, or the maximum amount of energy of position which the system can acquire, is thus dependent on the length of the pendulum arm. These two factors, then, the mass and the length of the pendulum arm, are simply properties of this pendulum machine, properties by which its energy compass is restricted. Let us now examine these limiting factors more minutely.
It is obvious that energy could readily be applied to the pendulum system in such a degree as to cause it to rotate with considerable angular velocity about the axis of suspension. Now the motion of the pendulum mass in the lines of the gravitation field, although productive of the same transformation process, differs from that of a body moving vertically upward in that, while the latter has a linear movement, the former is constrained into a circular path. This restraint is imposed in virtue of the cohesive properties of the material of the pendulum arm, and it is the presence of this restraining influence that really distinguishes the pendulum machine from the machine in which the moving mass is constrained by gravity alone (§ 20). It has been shown that the energy capacity of a body projected vertically against gravity is limited by its mass only; the energy capacity of the pendulum machine may be likewise limited by its mass, but the additional restraining factor of cohesion also imposes another limit. In the course of rotation, energy is stored in the material of the pendulum against the internal forces of cohesion. The action is simply that of what is usually termed centrifugal force. As the velocity increases, the pendulum arm lengthens correspondingly until the elastic limit of the material in tension is reached. At this point, the pendulum may be said to have reached the maximum length at which it can operate in that reversible process of transformation in which energy of motion is converted into energy of position. The amount of energy which would now be working in that process may be termed the limiting energy for reversibility. This limiting energy is the absolute maximum amount of energy which can operate in the reversible cycle. It is coincident with the maximum length of the pendulum arm in distortion. When the stress in the material of that arm reaches the elastic limit, it is clear that the transformation against cohesion will also have attained its limiting value for reversibility. This transformation, if the velocity of the pendulum is constant, is of the nature of a storage of energy. So long as the velocity is constant the energy stored is constant. If the elastic limiting stress of the material has not been exceeded, this energy—neglecting certain minor processes (§§ 15, 29)—will be returned in its original form as the velocity decreases. If, however, the material be stressed beyond its elastic powers, the excess energy applied will simply lead to permanent distortion or disruption of the pendulum arm, and to a complete breakdown and change in the character of the machine and the associated energy processes (§ 5). The physical properties of the material thus limit the energy capacity of the machine. This limiting feature, as already indicated, is not peculiar to the pendulum machine alone. Every energy process embodied in a material machine is limited in a similar fashion by the peculiar properties of the acting materials. Every reversible process is carried out within limits thus clearly defined. Nature presents no exception to this rule, no example of a reversible energy system on which energy may be impressed in unlimited amount. On the contrary, all the evidence points to limitation of the strictest order in such processes.
