NECESSITY OF CLEAR MECHANICAL CONCEPTIONS.

By WILLIAM D. MARKS,

Whitney Professor of Dynamical Engineering, University of Pennsylvania.

The nomenclature of mechanical conceptions is a matter of common consent, and could soon be reduced to approximate uniformity were the majority of writers upon mechanics sufficiently painstaking and clear in their use of terms, and would they bear in mind that a clear physical conception of the meaning of terms used is of vastly greater importance than any subsequent display of skill and ingenuity in the mathematical manipulation of the symbolical expressions for them.

Every term used in mechanics should convey to the mind a distinct physical conception, capable of being expressed in intelligible language without recourse to symbolical notation; and until this fact is recognized and acted upon, mechanics will ever be a dreaded study to those who are forced to take it up, saving that small proportion of students thoughtful and patient enough to elaborate their own conceptions by careful decomposition and isolation of the elements of the symbolical expressions which are taken for the foundation stones of an elaborate mathematical structure.

None who are engaged in teaching can have failed to perceive the stupefying effects of a course of symbolical reasoning unaccompanied by any attempt to materialize the meaning of the expressions deduced, or have not noted the injury of a naturally. clear intellect in the attempt to memorize a mass of partially apprehended formulæ.

Great mathematical acquirements do not seem to be an absolute necessity, since discoveries in natural philosophy seem to point out new and appropriate methods of quantitative treatment rather than to render available the labors of the pure mathematicians.

Our knowledge of mathematics does but enable us to weigh and measure our results. Every new mechanical problem should first be analyzed by means of a course of abstract reasoning before being quantitatively analyzed with the aid of mathematics, just as a cautious chemist precedes a quantitative analysis by qualitatively determining the nature and ingredients of the substance under consideration.

Clear ideas of the meanings of mechanical terms are imperatively

required as a first condition of success in a preliminary analysis of any problem, and any attempt at quantitative analysis will more probably lead to error than truth unless this preliminary analysis be complete.

That the writer may not appear to have "set up a man of straw for the pleasure of demolishing him, he will instance a few cases occurring in the works of the abler writers upon mechanics, passing over without notice the too apparent evasions and misconceptions of a host of writers of so-called "elementary mechanics."

Prof. Wm. Whewell, who is probably the clearest writer in the English language upon mechanics, constantly uses the terms force and and pressure interchangeably. Pressure refers rather to force distributed over a considerable surface, as in the case of water, the atmosphere, etc.

Prof. Rankine calls the moment of inertia of a revolving body the weight multiplied by the square of the radius of gyration; this expression is not the moment of inertia, but only the measure of the moment of inertia.

Prof. Tyndall defines heat as "a mode of motion"; it is really a form of work. Possibly this apparent error is a wilful misstatement, made with a design to convey to his readers an approximate idea of what he did not believe them capable of conceiving fully. It certainly is either an error or a concession to ignorance, which has done much harm.

It is very much easier to criticize defects than to remedy them, and the author, in offering the following verbal definitions, does not feel that he has made himself as clear as he could have wished to be.

He trusts, however, that they will serve the purpose of showing more clearly the meaning of the usual terms of mechanics, and their relations to each other, giving in the present form of successive aphorisms a connected view of the whole field of mechanics.

In order to be perfectly clear, and establish a complete understanding between our readers and ourselves, we will have to repeat the most elementary ideas, because the terms having the more complex meanings will demand the most precise accord as to the meaning of the elementary terms to which they will be reduced. We may, then, be pardoned for the repetition of definitions with which all are assumed to be familiar.

Dynamics may be separated into two studies-kinematics and

statics. When these two are considered in conjunction we have dynamics.

The careful isolation of these two branches of dynamics, and their separate study, will add much to the power of apprehension of the student when he comes to consider them conjointly.

Kinematics evades all questions of force, and in it we confine ourselves entirely to the consideration of the path, velocity and direction of motion.

Motion can best be defined as a change of position, and in many cases the velocity of this change is a matter of indifference, so that the path and direction of motion only receive our consideration. If velocity is taken into consideration we introduce the element of time, since the velocity of a point is the distance which is [or would be if the velocity was constant] passed over in whatever unit of time is used as a standard. One second is the usual standard.

We can then say in uniform motion the space described in any time is equal to the product of the velocity and the time.

When the velocity is not constant it can no longer be measured by the quotient of the space by the time, since these quotients will be different for different periods, and in variable velocities we measure the velocity, at any instant, by the space which would have been passed over in the succeeding second had the velocity been rendered constant at that instant.

Angular velocity, which is used to compare the speeds of rotation of bodies around their axes, can also be constant or variable; it is the velocity, in a circular path, of a point which is at a radial distance equal to unity from the axis of rotation of any rotating body; or, if the axis does not pass through the body, it is the velocity, in a circular path, of a point situated at a distance from the axis equal to unity, and in a an assumed line joining the axis and the body revolving around it. It can always be obtained by dividing the curvilinear velocity of any point in a rotating or revolving body by its radial distance from the axis.

Equipped with these few fundamental conceptions, we have all that are necessary to the study of kinematics, this word being used in its most limited sense.

Statics, on the other hand, evades all questions of motion, and in it we confine ourselves to the study of forces at rest.

What force really is we will probably never know until we learn

the ultimate nature of matter; we do know, however, that whatever tends to produce motion, or actually produces motion in bodies at rest; or brings or tends to bring a moving body to rest, or to change the direction of a moving body, is called force. We measure force by its intensity, in pounds, and limit it by its direction and point of application.

In terrestrial mechanics, to which we limit ourselves in this paper, gravity is the force which attracts all bodies to the surface of the earth in a vertical line; if allowed to act on a free body in a vacuum it will produce a velocity of about 32.2 ft. at the end of one second, during which time the body will have fallen a distance of 16.1 feet.

In order to define the centre of gravity of a body we will have to precede it by the definition of the statical moment of a force which is the intensity of that force multiplied by its perpendicular distance from the point around which it tends to, or actually does produce motion.

We can now define the centre of gravity of a body as that point which, if supported, leaving the body free to rotate in any direction, would balance all the moments of the forces of the molecules of the body due to the force of gravity; the body would not have any tendency to turn about this point, at which the total force of gravity acting upon the body may be assumed to be concentrated.

The well-known theorems of the parallelogram and parallelopipedon of forces form the basis for the statical treatment of forces which has received an enormous development both analytically and graphically.

We come now to dynamics, which is the study of combined force and motion, that is, of work; or if time in which the work is accomplished is included in the consideration, of power.

We can say that work equals force multiplied by the space passed over during the action of the force, and as the unit of space usually assumed is a foot and the unit of force a pound, we measure work in foot-pounds. A foot-pound is the amount of work done in raising a weight of one pound one foot, or in exerting a force of one pound through a distance of one foot in any direction. Work is considered independently of the time in which it is accomplished.

Power is work considered with respect to the time in which it is accomplished; as, for instance, a horse-power, which represents 33,000

foot-pounds of work done in one minute or 550 foot-pounds of work done in one second.

The many terms used to express the idea of force and motion combined can all be seen, by a little thought, to be synonymous with work. Power is often used incorrectly for force when the lever and screw are being discussed.

The weight of a body is the measure of the intensity of the force of gravity acting upon it. In treatises on mechanics it is made equal to the product of its mass by its velocity at the end of one second (32.2 feet) under the action of gravity.

In order to clearly grasp the meaning of this last sentence we must know what mass is. The mass of a body is usually stated to be the quantity of matter in it, and it will at once be perceived that the hypothesis is placed that differences in quantity of matter make proportional differences in the weight, which may or may not be true. We have no means of proving that a volume of iron which weighs 7.2 times as much as the same volume of water contains 7.2 times as much matter.

The fact is that mass means the intensity of the force of gravity divided by the velocity due to the force of gravity at the end of one second, and is a constant ratio at all points on the surface of the earth. The great convenience of this ratio for the purposes of the mechanic will be seen when we recollect that in dynamics the intensity of a force is measured by the velocity which it will produce in one second, and if we multiply this ratio (which is the mass) by the velocity which is observed, we have the intensity of the acting force in pounds. This leads us at once to the momentum of a body, which is the intensity of a constant force which has been (or should have been to produce the same velocity) acting upon it for one second, it is equal to its mass multiplied by its velocity, in feet per second.

If a moving body be brought to rest in one second, the momentum is the intensity of the constant force which must be exerted through a space equal to one-half the velocity of the moving body.

In speaking of momentum, unity of time is always assumed as one of the conditions; thus the weight of a body equals its momentum when gravity is the force acting upon it.

The distinction between momentum and the really acting force in many cases which occur must be sharply drawn. If the acting force be constant, it will add equal increments of velocity in each unit of