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Then, as before,
(mean of OP2)=01? + (mean of IP); since the mean of IQ is still zero, IQ being the distance of a point of the system from the plane through I perpendicular to 01.
198. If the masses of the points be unequal, it is easy to see (as in § 195) that the first of these theorems becomes
The sum of the squares of the distances of the parts of a system from any point, each multiplied by the mass of that part, exceeds the corresponding quantity for the centre of inertia by the product of the square of the distance of the point from the centre of inertia, by the whole mass of the system.
Also, the sum of the products of the mass of each part of a system by the square of its distance from any axis is called the Moment of Inertia of the system about this axis; and the second proposition above is equivalent to
The moment of inertia of a system about any axis is equal to the moment of inertia about a parallel axis through the centre of inertia, 1, together with the moment of inertia, about the first axis, of the whole mass supposed condensed at I.
199. The Moment of any physical agency is the numerical measure of its importance. Thus, the moment of inertia of a body round an axis (§ 198) means the importance of its inertia relatively to rotation round that axis. Again, the moment of a force round a point or round a line ($ 46), signifies the measure of its importance as regards producing or balancing rotation round that point or round that line.
It is often convenient to represent the moment of a force by a line numerically equal to it, drawn through the vertex of the triangle representing its magnitude, perpendicular to its plane, through the front of a watch held in the plane with its centre at the point, and facing so that the force tends to turn round this point in a direction opposite to the hands. The moment of a force round any axis is the moment of its component in any plane perpendicular to the axis, round the point in which the plane is cut by the axis. Here we imagine the force resolved into two components, one parallel to the axis, which is ineffective so far as rotation round the axis is concerned; the other perpendicular to the axis (that is to say, having its line in any plane perpendicular to the axis). This latter component may be called the effective component of the force, with reference to rotation round the axis. And its moment round the axis may be defined as its moment round the nearest point of the axis, which is equivalent to the preceding definition. It is clear that the moment of a force round any axis, is equal to the area of the projection on any plane perpendicular to the axis, of the figure representing its moment round any point of the axis.
200. [The projection of an area, plane or curved, on any plane, is the area included in the projection of its bounding line.
If we imagine an area divided into any number of parts, the projections of these parts on any plane make up the projection of the whole.
But in this statement it must be understood that the areas of partial projections are to be reckoned as positive if particular sides, which, for brevity, we may call the outside of the projected area and the front of the plane of projection, face the same way, and negative if they face oppositely.
Of course if the projected surface, or any part of it, be a plane area at right angles to the plane of projection, the projection vanishes. The projections of any two shells having a common edge, on any plane, are equal. The projection of a closed surface (or a shell with evanescent edge), on any plane, is nothing.
Equal areas in one plane, or in parallel planes, have equal projections on any plane, whatever may be their figures.
Hence the projection of any plane figure, or of any shell edged by a plane figure, on another plane, is equal to its area, multipled by the cosine of the angle at which its plane is inclined to the plane of projection. This angle is acute or obtuse, according as the outside of the projected area, and the front of the plane of projection, face on the whole towards the same parts, or oppositely. Hence lines representing, as above described, moments about a point in different planes, are to be compounded as forces are. analogous theorem in § 107.]
201. A Couple is a pair of equal forces acting in dissimilar directions in parallel lines. The Moment of a couple is the sum of the moments of its forces about any point in their plane, and is therefore equal to the product of either force into the shortest distance between their directions. This distance is called the Arm of the couple.
The Axis of a Couple is a line drawn from any chosen point of reference perpendicular to the plane of the couple, of such magnitude and in such direction as to represent the magnitude of the moment, and to indicate the direction in which the couple tends to turn. The most convenient rule for fulfilling the latter condition is this :—Hold a watch with its centre at the point of reference, and with its plane parallel to the plane of the couple. Then, according as the motion of the hands is contrary to, or along with the direction in which the couple tends to turn, draw the axis of the couple through the face or through the back of the watch. It will be found that a couple is completely represented by its axis, and that couples are to be resolved and compounded by the same geometrical constructions performed with reference to their axes as forces or velocities, with reference to the lines directly representing them.
202. By introducing in the definition of moment of velocity ($ 46) the mass of the moving body as a factor, we have an important element of dynamical science, the Moment of Momentum. The laws of composition and resolution are the same as those already explained.
203. [If the point of application of a force be displaced through a small space, the resolved part of the displacement in the direction of the force has been called its Virtual Velocity. This is positive or
negative according as the virtual velocity is in the same, or in the opposite, direction to that of the force.
The product of the force, into the virtual velocity of its point of application, has been called the Virtual Moment of the force. These terms we have introduced since they stand in the history and developments of the science; but, as we shall show further on, they arë inferior substitutes for a far more useful set of ideas clearly laid down by Newton.]
204. A force is said to do work if its place of application has a positive component motion in its direction; and the work done by it is measured by the product of its amount into this component motion.
Generally, unit of work is done by unit force acting through unit space. In lifting coals from a pit, the amount of work done is proportional to the weight of the coals lifted; that is, to the force overcome in raising them; and also to the height through which they are raised. The unit for the measurement of work adopted in practice by British engineers, is that required to overcome a force equal to the weight of a pound through the space of a foot; and is called a FootPound. (See § 185.)
In purely scientific measurements, the unit of work is not the footpound, but the kinetic unit force ($ 190) acting through unit of space. Thus, for example, as we shall show further on, this unit is adopted in measuring the work done by an electric current, the units for electric and magnetic measurements being founded upon the kinetic unit force.
If the weight be raised obliquely, as, for instance, along a smooth inclined plane, the space through which the force has to be overcome is increased in the ratio of the length to the height of the plane; but the force to be overcome is not the whole weight, but only the resolved part of the weight parallel to the plane; and this is less than the weight in the ratio of the height of the plane to its length. By multiplying these two expressions together, we find, as we might expect, that the amount of work required is unchanged by the substitution of the oblique for the vertical path.
205. Generally, for any force, the work done during an indefinitely small displacement of the point of application is the virtual moment of the force ($ 203), or is the product of the resolved part of the force in the direction of the displacement into the displacement.
From this it appears, that if the motion of the point of application be always perpendicular to the direction in which a force acts, such a force does no work. Thus the mutual normal pressure between a fixed and moving body, the tension of the cord to which a pendulum bob is attached, or the attraction of the sun on a planet if the planet describe a circle with the sun in the centre, are all instances in which no work is done by the force.
206. The work done by a force, or by a couple, upon a body turning about an axis, is the product of the moment of either into the angle (in circular measure) through which the body acted on turns, if
the moment remains the same in all positions of the body. If the moment be variable, the above assertion is only true for indefinitely small displacements, but may be made accurate by employing the proper average moment of the force or of the couple. The proof is obvious.
207. Work done on a body by a force always wn by a corresponding increase of vis viva, or kinetic energy, if no other forces act on the body which can do work or have work done against them. If work be done against any forces, the increase of kinetic energy is less than in the former case by the amount of work so done. In virtue of this, however, the body possesses an equivalent in the form of Potential Energy ($ 239), if its physical conditions are such that these forces will act equally, and in the same directions, if the motion of the system is reversed. Thus there may be no change of kinetic energy produced, and the work done may be wholly stored up as potential energy.
Thus a weight requires work to raise it to a height, a spring requires work to bend it, air requires work to compress it, etc.; but a raised weight, a bent spring, compressed air, etc., are stores of energy which can be made use of at pleasure.
208. In what precedes we have given some of Newton's Definitiones nearly in his own words; others have been enunciated in a form more suitable to modern methods; and some terms have been introduced which were invented subsequent to the publication of the Principia. But the Axiomata, sive Leges Motús, to which we now proceed, are given in Newton's own words. The two centuries which have nearly elapsed since he first gave them have not shown a necessity for any addition or modification. The first two, indeed, were discovered by Galileo: and the third, in some of its many forms, was known to Hooke, Huyghens, Wallis, Wren, and others, before the publication of the Principia. Of late there has been a tendency to divide the second law into two, called respectively the second and third, and to ignore the third entirely, though using it directly in every dynamical problem; but all who have done so have been forced indirectly to acknowledge the incompleteness of their substitute for Newton's system, by introducing as an axiom what is called D'Alembert's principle, which is really a deduction from Newton's rejected third law. Newton's own interpretation of his third law directly points out not only D'Alembert's principle, but also the modern principles of Work and Energy.
209. An Axiom is a proposition, the truth of which must be admitted as soon as the terms in which it is expressed are clearly understood. And, as we shall show in our chapter on · Experience,' physical axioms are axiomatic to those who have sufficient knowledge of physical phenomena to enable them to understand perfectly what is asserted by them. Without further remark we shall give Newton's Three Laws; it being remembered that, as the properties of matter might have been such as to render a totally different set of laws axiomatic, these laws must be considered as resting on convictions drawn from observation and experiment, not on intuitive perception,
210. LEX I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus illud à viribus impressis cogitur statum suum mutare.
Every body continues in its state of rest or of uniform motion in a straight line, except in so far as it may be compelled by impressed forces to change that state.
211. The meaning of the term Rest, in physical science, cannot be absolutely defined, inasmuch as absolute rest nowhere exists in nature. If the universe of matter were finite, its centre of inertia might fairly be considered as absolutely at rest; or it might be imagined to be moving with any uniform velocity in any direction whatever through
But it is remarkable that the first law of motion enables us (§ 215, below) to explain what may be called directional rest. Also, as will be seen farther on, a perfectly smooth spherical body, made up of concentric shells, each of uniform material and density throughout, if made to revolve about an axis, will, in spite of impressed forces, revolve with uniform angular velocity, and will maintain its axis of revolution in an absolutely fixed direction. Or, as will soon be shown (§ 233), the plane in which the moment of momentum of the universe (if finite) round its centre of inertia is the greatest, which is clearly determinable from the actual motions at any instant, is fixed in direction in space.
212. We may logically convert the assertion of the first law of motion as to velocity into the following statements :
The times during which any particular body, not compelled by force to alter the speed of its motion, passes through equal spaces, are equal. And, again-Every other body in the universe, not compelled by force to alter the speed of its motion, moves over equal spaces in successive intervals, during which the particular chosen body moves over equal spaces.
213. The first part merely expresses the convention universally adopted for the measurement of Time. The earth in its rotation about its axis, presents us with a case of motion in which the condition of not being compelled by force to alter its speed, is more nearly fulfilled than in any other which we can easily or accurately observe. And the numerical measurement of time practically rests on defining equal intervals of time, as times during which the earth turns through equal angles. This is, of course, a mere convention, and not a law of nature; and, as we now see it, is a part of Newton's first law.
214. The remainder of the law is not a convention, but a great truth of nature, which we may illustrate by referring to small and trivial cases as well as to the grandest phenomena we can conceive.
A curling-stone, projected along a horizontal surface of ice, travels equal distances, except in so far as it is retarded by friction and by the resistance of the air, in successive intervals of time during which the earth turns through equal angles. The sun moves through equal portions of interstellar space in times during which the earth turns