A Treatise on Mechanics — Reading Notes

avity will move and have the same velocity as if the whole mass were there concentrated and received the impelling forces.

(179.) These general properties, which are entirely independent of gravity, render the “centre of gravity” an inadequate title for this important point. Some physical writers have, consequently, called it the “centre of inertia.” The “centre of gravity,” however, is the name by which it is still generally designated.

THE MECHANICAL PROPERTIES OF AN AXIS.

(180.) When a body has a motion of rotation, the line round which it revolves is called an _axis_. Every point of the body must in this case move in a circle, whose centre lies in the axis, and whose radius is the distance of the point from the axis. Sometimes while the body revolves, the axis itself is moveable, and not unfrequently in a state of actual motion. The motions of the earth and planets, or that of a common spinning-top, are examples of this. The cases, however, which will be considered in the present chapter, are chiefly those in which the axis is immovable, or at least where its motion has no relation to the phenomena under investigation. Instances of this are so frequent and obvious, that it seems scarcely necessary to particularise them. Wheel-work of every description, the moving parts of watches and clocks, turning lathes, mill-work, doors and lids on hinges, are all obvious examples. In tools or other instruments which work on joints or pivots, such as scissors, shears, pincers, although the joint or pivot be not absolutely fixed, it is to be considered so in reference to the mechanical effect.

In some cases, as in most of the wheels of watches and clocks, fly-wheels and chucks of the turning lathe, and the arms of wind-mills, the body turns continually in the same direction, and each of its points traverses a complete circle during every revolution of the body round its axis. In other instances the motion is alternate or reciprocating, its direction being at intervals reversed. Such is the case in pendulums of clocks, balance-wheels of chronometers, the treddle of the lathe, doors and lids on hinges, scissors, shears, pincers, &c. When the alternation is constant and regular, it is called _oscillation_ or _vibration_, as in pendulums and balance-wheels.

(181.) To explain the properties of an axis of rotation it will be necessary to consider the different kinds of forces to the action of which a body moveable on such an axis may be submitted, to show how this action depends on their several quantities and directions, to distinguish the cases in which the forces neutralise each other and mutually equilibrate from those in which motion ensues, to determine the effect which the axis suffers, and, in the cases where motion is produced, to estimate the effects of those centrifugal forces (137.) which are created by the mass of the body whirling round the axis.

Forces in general have been distinguished by the duration of their action into instantaneous and continued forces. The effect of an instantaneous force is produced in an infinitely short time. If the body which sustains such an action be previously quiescent and free, it will move with a uniform velocity in the direction of the impressed force. (93.) If, on the other hand, the body be not free, but so restrained that the impulse cannot put it in motion, then the fixed points or lines which resist the motion sustain a corresponding shock at the moment of the impulse. This effect, which is called _percussion_, is, like the force which causes it, instantaneous.

A continued force produces a continued effect. If the body be free and previously quiescent, this effect is a continual increase of velocity. If the body be so restrained that the applied force cannot put it in motion, the effect is a continued pressure on the points or lines which sustain it. (94.)

It may happen, however, that although the body be not absolutely free to move in obedience to the force applied to it, yet still it may not be altogether so restrained as to resist the effect of that force and remain at rest. If the point at which a force is applied be free to move in a certain direction not coinciding with that of the applied force, that force will be resolved into two elements; one of which is in the direction in which the point is free to move, and the other at right angles to that direction. The point will move in obedience to the former element, and the latter will produce percussion or pressure on the points or lines which restrain the body. In fact, in such cases the resistance offered by the circumstances which confine the motion of the body modifies the motion which it receives, and as every change of motion must be the consequence of a force applied (44.), the fixed points or lines which offer the resistance must suffer a corresponding effect.

It may happen that the forces impressed on the body, whether they be continued or instantaneous, are such as, were it free, would communicate to it a motion which the circumstances which restrain it do not forbid it to receive. In such a case the fixed points or lines which restrain the body sustain no force, and the phenomena will be the same in all respects as if these points or lines were not fixed.

It will be easy to apply these general reflections to the case in which a solid body is moveable on a fixed axis. Such a body is susceptible of no motion except one of rotation on that axis. If it be submitted to the action of instantaneous forces, one or other of the following effects must ensue. 1. The axis may resist the forces, and prevent any motion. 2. The axis may modify the effect of the forces sustaining a corresponding percussion, and the body receiving a motion of rotation. 3. The forces applied may be such as would cause the body to spin round the axis even were it not fixed, in which case the body will receive a motion of rotation, but the axis will suffer no percussion.

What has been just observed of the effect of instantaneous forces is likewise applicable to continued ones. 1. The axis may entirely resist the effect of such forces, in which case it will suffer a pressure which may be estimated by the rules for the composition of force. 2. It may modify the effect of the applied forces, in which case it must also sustain a pressure, and the body must receive a motion of rotation which is subject to constant variation, owing to the incessant action of the forces. 3. The forces may be such as would communicate to the body the same rotatory motion if the axis were not fixed. In this case the forces will produce no pressure on the axis.

The impressed forces are not the only causes which affect the axis of a body during the phenomenon of rotation. This species of motion calls into action other forces depending on the inertia of the mass, which produce effects upon the axis, and which play a prominent part in the theory of rotation. While the body revolves on its axis, the component particles of its mass move in circles, the centres of which are placed in the axis. The radius of the circle in which each particle moves is the line drawn from that particle perpendicular to the axis. It has been already proved that a particle of matter, moving round a centre, is attended with a centrifugal force proportionate to the radius of the circle in which it moves and to the square of its angular velocity. When a solid body revolves on its axis, all its parts are whirled round together, each performing a complete revolution in the same time. The angular velocity is consequently the same for all, and the difference of the centrifugal forces of different particles must entirely depend upon their distances from the axis. The tendency of each particle to fly from the axis, arising from the centrifugal force, is resisted by the cohesion of the parts of the mass, and in general this tendency is expended in exciting a pressure or strain upon the axis. It ought to be recollected, however, that this pressure or strain is altogether different from that already mentioned, and produced by the forces which give motion to the body. The latter depends entirely upon the quantity and directions of the applied forces in relation to the axis: the former depends on the figure and density of the body, and the velocity of its motion.

These very complex effects render a simple and elementary exposition of the mechanical properties of a fixed axis a matter of considerable difficulty. Indeed, the complete mathematical development of this theory long eluded the skill of the most acute geometers, and it was only at a comparatively late period that it yielded to the searching analysis of modern science.

(182.) To commence with the most simple case, we shall consider the body as submitted to the action of a single force. The effect of this force will vary according to the relation of its direction to that of the axis. There are two ways in which a body may be conceived to be moveable around an axis. 1. By having pivots at two points which rest in sockets, so that when the body is moved it must revolve round the right line joining the pivots as an axis. 2. A thin cylindrical rod may pass through the body, on which it may turn in the same manner as a wheel upon its axle.

If the force be applied to the body in the direction of the axis, it is evident that no motion can ensue, and the effect produced will be a pressure on that pivot towards which the force is directed. If in this case the body revolved on a cylindrical rod, the tendency of the force would be to make it slide along the rod without revolving round it.

Let us next suppose the force to be applied not in the direction of the axis itself, but parallel to it. Let A B, _fig. 70._, be the axis, and let C D be the direction of the force applied. The pivots being supposed to be at A and B, draw A G and B F perpendicular to A B. The force C D will be equivalent to three forces, one acting from B towards A, equal in quantity to the force C D. This force will evidently produce a corresponding pressure on the pivot A. The other two forces will act in the directions A G and B F, and will have respectively to the force C D the same proportion as A E has to A B. Such will be the mechanical effect of a force C D parallel to the axis. And as these effects are all directed on the pivots, no motion can ensue.

If the body revolve on a cylindrical rod, the forces A G and B F would produce a strain upon the axis, while the third force in the direction B A would have a tendency to make the body slide along it.

(183.) If the force applied to the body be directed upon the axis, and at right angles to it, no motion can be produced. In this case, if the body be supported by pivots at A and B, the force K L, perpendicular to the line A B, will be distributed between the pivots, producing a pressure on each proportional to its distance from the other. The pressure on A having to the pressure on B the same proportion as L B has to L A.

If the force K H be directed obliquely to the axis, it will be equivalent to two forces (76.), one K L perpendicular to the axis, and the other K M parallel to it. The effect of each of these may be investigated as in the preceding cases.

In all these observations the body has been supposed to be submitted to the action of one force only. If several forces act upon it, the direction of each of them crossing the axis either perpendicularly or obliquely, or taking the direction of the axis or any parallel direction, their effects may be similarly investigated. In the same manner we may determine the effects of any number of forces whose combined results are mechanically equivalent to forces which either intersect the axis or are parallel to it.

(184.) If any force be applied whose direction lies in a plane oblique to the axis, it can always be resolved into two elements (76.), one of which is parallel to the axis, and the other in a plane perpendicular to it. The effect of the former has been already determined, and therefore we shall at present confine our attention to the latter.

Suppose the axis to be perpendicular to the paper, and to pass through the point G, _fig. 71._ and let A B C be a section of the body. It will be convenient to consider the section vertical and the axis horizontal, omitting, however, any notice of the effect of the weight of the body.

Let a weight W be suspended by a cord Q W from any point Q. This weight will evidently have a tendency to turn the body round in the direction A B C. Let another cord be attached to any other point P, and, being carried over a wheel R, let a dish S be attached to it, and let fine sand be poured into this dish until the tendency of S to turn the body round the axis in the direction of C B A balances the opposite tendency of W. Let the weights of W and S be then exactly ascertained, and also let the distances G I and G H of the cords from the axis be exactly measured. It will be found that, if the number of ounces in the weight S be multiplied by the number of inches in G H, and also the number of ounces in W by the number of inches in G I, equal products will be obtained. This experiment may be varied by varying the position of the wheel R, and thereby changing the direction of the string P R, in which cases it will be always found necessary to vary the weight of S in such a manner, that when the number of ounces in it is multiplied by the number of inches in the distance of the string from the axis, the product obtained shall be equal to that of the weight W by the distance G I. We have here used ounces and inches as the measures of weight and distance; but it is obvious that any other measures would be equally applicable.

From what has been just stated it follows, that the energy of the weight of S to move the body on its axis, does not depend alone upon the actual amount of that weight, but also upon the distance of the string from the axis. If, while the position of the string remains unaltered, the weight of S be increased or diminished, the resisting weight W must be increased or diminished in the same proportion. But if, while the weight of S remains unaltered, the distance of the string P R from the axis G be increased or diminished, it will be found necessary to increase or diminish the resisting weight W in exactly the same proportion. It therefore appears that the increase or diminution of the distance of the direction of a force from the axis has the same effect upon its power to give rotation as a similar increase or diminution of the force itself. The power of a force to produce rotation is, therefore, accurately estimated, not by the force alone, but by the product found by multiplying the force by the distance of its direction from the axis. It is frequently necessary in mechanical science to refer to this power of a force, and, accordingly, the product just mentioned has received a particular denomination. It is called the _moment_ of the force round the axis.

(185.) The distance of the direction of a force from the axis is sometimes called the _leverage_ of the force. The _moment_ of a force is therefore found by multiplying the force by its leverage, and the energy of a given force to turn a body round an axis is proportional to the leverage of that force.

From all that has been observed it may easily be inferred that, if several forces affect a body moveable on an axis, having tendencies to turn it in different directions, they will mutually neutralise each other and produce equilibrium, if the sum of the moments of those forces which tend to turn the body in one direction be equal to the sum of the moments of those which tend to turn it in the opposite direction. Thus, if the forces A, B, C, ... tend to turn the body from right to left, and the distances of their directions from the axis be _a_, _b_, _c_, ... and the forces A′, B′, C′, ... tend to move it from left to right, and the distances of their directions from the axis be _a′_, _b′_, _c′_, ...; then these forces will produce equilibrium, if the products found by multiplying the ounces in A, B, C, ... respectively by the inches in _a_, _b_, _c_, ... when added together be equal to the products found by multiplying the ounces in A′, B′, C′, ... by the inches in _a′_, _b′_, _c′_, ... respectively when added together. But if either of these sets of products when added together exceed the other, the corresponding set of forces will prevail, and the body will revolve on its axis.

(186.) When a body receives an impulse in a direction perpendicular to the axis, but not crossing it, a uniform rotatory motion is produced. The velocity of this motion depends on the force of the impulse, the distance of the direction of the impulse from the axis, and the manner in which the mass of the body is distributed round the axis. It is to be considered that the whole force of the impulse is shared amongst the various parts of the mass, and is transmitted to them from the point where the impulse is applied by reason of the cohesion and tenacity of the parts, and the impossibility of one part yielding to a force without carrying all the other parts with it. The force applied acts upon those particles nearer to the axis than its own direction under advantageous circumstances; for, according to what has been already explained, their power to resist the effect of the applied force is small in the same proportion with their distance. On the other hand, the applied force acts upon particles of the mass, at a greater distance than its own direction, under circumstances proportionably disadvantageous; for their resistance to the applied force is great in proportion to their distances from the axis.

Let C D, _fig. 72._, be a section of the body made by a plane passing through the axis A B. Suppose the impulse to be applied at P, perpendicular to this plane, and at the distance P O from the axis. The effect of the impulse being distributed through the mass will cause the body to revolve on A B, with a uniform velocity. There is a certain point G, at which, if the whole mass were concentrated, it would receive from the impulse the same velocity round the axis. The distance O G is called the _radius of gyration_ of the axis A B, and the point G is called the _centre of gyration_ relatively to that axis. The effect of the impulse upon the mass concentrated at G is great in exactly the same proportion as O G is small. This easily follows from the property of moments which has been already explained; from whence it may be inferred, that the greater the radius of gyration is, the less will be the velocity which the body will receive from a given impulse.