Chapter V: Part 5

Since P F represents the force of gravity, that is, the force with which the body would descend freely in the vertical direction, and P C the force with which it moves down the plane, it follows that a body would fall freely in the vertical direction from P to F in the same time as on the plane it would move from P to C. In this manner, therefore, when the height through which a body would fall vertically is known, the space through which it would descend in the same time down any given inclined plane may be immediately determined. For let A B, _fig. 25._, be the given inclined plane, and let P F be the space through which the body would fall in one second. From F draw F C perpendicular to the plane, and the space P C is that through which the body P will fall in one second on the plane.

(133.) As the angle B A H, which measures the elevation of the plane, is increased, the obliquity of the vertical direction P F with the plane is also increased. Consequently, according to what has been proved (130), it follows, that as the elevation of the plane is increased, the force which urges the body down the plane is also increased, and as the elevation is diminished, the force suffers a corresponding diminution. The two extreme cases are, 1. When the plane is raised until it becomes perpendicular, in which case the weight is permitted to fall freely, without exerting any pressure upon the plane; and, 2. When the plane is depressed until it becomes horizontal, in which case the whole weight is supported, and there is no motion.

From these circumstances it follows, that by means of an inclined plane we can obtain an uniformly-accelerating force of any magnitude less than that of gravity.

We have here omitted, and shall for the present in every instance omit, the effects of _friction_, by which the motion down the plane is retarded. Having first investigated the mechanical properties of bodies supposed to be free from friction, we shall consider friction separately, and show how the present results are modified by it.

(134.) The accelerating forces on different inclined planes may be compared by the principle explained in (131). Let _figs. 25._ and _26._ be two inclined planes, and take the lines P F in each figure equal, both expressing the force of gravity, then P C will be the force which in each case urges the body down the plane.

As the force down an inclined plane is less than that which urges a body falling freely in the vertical direction, the space through which the body must fall to attain a certain final velocity must be just so much greater as the accelerating force is less. On this principle we shall be able to determine the final velocity in descending through any space on a plane, compared with the final velocity attained in falling freely in the vertical direction. Suppose the body P, _fig. 27._, placed at the top of the plane, and from H draw the perpendicular H C. If B H represent the force of gravity, B C will represent the force down the plane (131). In order that the body moving down the plane shall have a final velocity equal to that of one which has fallen freely from B to H, it will be necessary that it should move from B down the plane, through a space which bears the same proportion to B H as B H does to B C. But since the triangle A B H is in all respects similar to H B C, only made upon a larger scale, the line A B bears the same proportion to B H as B H bears to B C. Hence, in falling on the inclined plane from B to A, the final velocity is the same as in falling freely from B to H.

It is evident that the same will be true at whatever level an horizontal line be drawn. Thus, if I K be horizontal, the final velocity in falling on the plane from B to I will be the same as the final velocity in falling freely from B to K.

(135.) The motion of a heavy body down a curve differs in an important respect from the motion down an inclined plane. Every part of the plane being equally inclined to the vertical direction, the effect of gravity in the direction of the plane is uniform; and, consequently, the phenomena obey all the established laws of uniformly-accelerated motion. If, however, we suppose the line B A, on which the body P descends, to be curved as in _fig. 28._, the obliquity of its direction at different parts, to the direction P F of gravity, will evidently vary. In the present instance, this obliquity is greater towards B and less towards A, and hence the part of the force of gravity which gives motion to the body is greater towards B than towards A (130). The force, therefore, which urges the body, instead of being uniform as in the inclined plane, is here gradually diminished. The rate of this diminution depends entirely on the nature of the curve, and can be deduced from the properties of the curve by mathematical reasoning. The details of such an investigation are not, however, of a sufficiently elementary character to allow of being introduced with advantage into this treatise. We must therefore limit ourselves to explain such of the results as may be necessary for the development of the other parts of the science.

(136.) When a heavy body is moved down an inclined plane by the force of gravity, the plane has been proved to sustain a pressure, arising from a certain part of the weight P D, _fig. 25._, which acts perpendicularly to the plane. This is also the case in moving down a curve such as B A, _fig. 28._ In this case, also, the whole weight is distributed between that part which is directed down the curve, and that which, being perpendicular to the curve, produces a pressure upon it. There is, however, another cause which produces pressure upon the curve, and which has no operation in the case of the inclined plane. By the property of inertia, when a body is put in motion in any direction, it must persevere in that direction, unless it be deflected from it by an efficient force. In the motion down an inclined plane the direction is never changed, and therefore by its inertia the falling body retains all the motion impressed upon it continually in the same direction; but when it descends upon a curve, its direction is constantly varying, and the resistance of the curve being the deflecting cause, the curve must sustain a pressure equal to that force, which would thus be capable of continually deflecting the body from the rectilinear path in which it would move in virtue of its inertia. This pressure entirely depends on the curvature of the path in which the body is constrained to move, and on its inertia, and is therefore altogether independent of the weight, and would, in fact, exist if the weight were without effect.

(137.) This pressure has been denominated _centrifugal force_, because it evinces a tendency of the moving body to _fly from_ the centre of the curve in which it is moved. Its quantity depends conjointly on the velocity of the motion and the curvature of the path through which the body is moved. As circles may be described with every degree of curvature, according to the length of the radius, or the distance from their circumference to their centre, it follows that, whatever be the curve in which the body moves, a circle can always be assigned which has the same curvature as is found at any proposed point of the given curve. Such a circle is called “the circle of curvature” at that point of the curve; and as all curves, except the circle, vary their degrees of curvature at different points, it follows that different parts of the same curve will have different circles of curvature. It is evident that the greater the radius of a circle is, the less is its curvature: thus the circle with the radius A B, _fig. 29._, is more curved than that whose radius is C D, and that in the exact proportion of the radius C D to the radius A B. The radius of the circle of curvature for any part of a curve is called “the radius of curvature” of that part.

(138.) The centrifugal pressure increases as the radius of curvature increases; but it also has a dependence on the velocity with which the moving body swings round the centre of the circle of curvature. This velocity is estimated either by the actual space through which the body moves, or by the _angular velocity_ of a line drawn from the centre of the circle to the moving body. That body carries one end of this line with it, while the other remains fixed at the centre. As this angular swing round the centre increases, the centrifugal pressure increases. To estimate the rate at which this pressure in general varies, it is necessary to multiply the square of the number expressing the angular velocity by that which expresses the radius of curvature, and the force increases in the same proportion as the product thus obtained.

(139.) We have observed that the same causes which produce pressure on a body restrained, will produce motion if the body be free. Accordingly, if a body be moved by any efficient cause in a curve, it will, by reason of the centrifugal force, _fly off_, and the moving force with which it will thus retreat from the centre round which it is whirled will be a measure of the centrifugal force. Upon this principle an apparatus called a _whirling table_ has been constructed, for the purpose of exhibiting experimental illustrations of the laws of centrifugal force. By this machine we are enabled to place any proposed weights at any given distances from centres round which they are whirled, either with the same angular velocity, or with velocities having a certain proportion. Threads attached to the whirling weights are carried to the centres round which they respectively revolve, and there, passing over pulleys, are connected with weights which may be varied at pleasure. When the whirling weights fly from their respective centres, by reason of the centrifugal force, they draw up the weights attached to the other ends of the threads, and the amount of the centrifugal force is estimated by the weight which it is capable of raising.

With this instrument the following experiments may be exhibited:--

Exp. 1. Equal weights whirled with the same velocity at equal distances from the centre raise the same weight, and therefore have the same centrifugal force.

Exp. 2. Equal weights whirled with the same angular velocity at distances from the centre in the proportion of one to two, will raise weights in the same proportion. Therefore the centrifugal forces are in that proportion.

Exp. 3. Equal weights whirled at equal distances with angular velocities which are as one to two, will raise weights as one to four, that is, as the squares of the angular velocities. Therefore the centrifugal forces are in that proportion.

Exp. 4. Equal weights whirled at distances which are as two to three, with angular velocities which are as one to two, will raise weights which are as two to twelve; that is, as the products of the distances two and three, and the squares one and four, of the angular velocities. Hence, the centrifugal forces are in this proportion.

The centrifugal force must also increase as the mass of the body moved increases; for, like attraction, each particle of the moving body is separately and equally affected by it. Hence a double mass, moving at the same distance, and with the same velocity, will have a double force. The following experiment verifies this:--

Exp. 5. If weights, which are as one to two, be whirled at equal distances with the same velocity, they will raise weights which are as one to two.

The law which governs centrifugal force may then be expressed in general symbols briefly thus:--

Let _c_ = the centrifugal force with which a weight of one lb. revolving in a circle in one second, the radius of which is one foot, would act on a string connecting it with the centre. The force with which it would act on a string, the length of which is R feet, would be _c_ × R; and if instead of revolving in one second it revolved in T seconds, the force would be

(_c_ × R)/T^2;

and if the revolving mass were W lbs. the force would be

C = (_c_ × W × R)/T^2.

This formula includes the entire theory of centrifugal force.

But it can be shown that the number expressed by _c_ is 1·226, and consequently

C = (1·226 × W × R)/T^2.

It is often more convenient to use the number of revolutions made in a given time than the time of one revolution. Let N then express the number of revolutions, or fraction of a revolution, made in one second, and we shall have

T = 1/N.

Therefore

C = 1·226 × W × R × N^2.

(140.) The consideration of centrifugal force proves, that if a body be observed to move in a curvilinear path, some efficient cause must exist which prevents it from flying off, and which compels it to revolve round the centre. If the body be connected with the centre by a thread, cord, or rod, then the effect of the centrifugal force is to give tension to the thread, cord, or rod. If an unyielding curved surface be placed on the convex side of the path, then the force will produce pressure on this surface. But if a body is observed to move in a curve without any visible material connection with its centre, and without any obstruction on the convex side of its path to resist its retreat, as is the case with the motions of the planets round the sun, and the satellites round the planets, it is usual to assign the cause to the attraction of the body which occupies the centre: in the present instance the sun is that body, and it is customary to say that the _attraction_ of the sun, neutralising the effects of the centrifugal force of the planets, _retains them_ in their orbits. We have elsewhere animadverted on the inaccurate and unphilosophical style of this phraseology, in which terms are admitted which intimate not only an unknown cause, but assign its seat, and intimate something of its nature. All that we are entitled to declare in this case is, that a motion is continually impressed upon the planet; that this motion is directed towards the sun; that it counteracts the centrifugal force; but from whence this motion proceeds, whether it be a virtue resident in the sun, or a property of the medium or space in which both sun and planets are placed, or whatever other influence may be its proximate cause, we are altogether ignorant.

* * * * *

(141.) Numerous examples of the effects of centrifugal force may be produced.

If a stone or other weight be placed in a sling, which is whirled round by the hand in a direction perpendicular to the ground, the stone will not fall out of the sling, even when it is at the top of its circuit, and, consequently, has no support beneath it. The centrifugal force, in this case, acting from the hand, which is the centre of rotation, is greater than the weight of the body, and therefore prevents its fall.

In like manner, a glass of water may be whirled so rapidly that even when the mouth of the glass is presented downwards, the water will still be retained in it by the centrifugal force.

If a bucket of water be suspended by a number of threads, and these threads be twisted by turning round the bucket many times in the same direction, on allowing the cords to untwist, the bucket will be whirled rapidly round, and the water will be observed to rise on its sides and sink at its centre, owing to the centrifugal force with which it is driven from the centre. This effect might be carried so far, that all the water would flow over and leave the bucket nearly empty.

(142.) A carriage, or horseman, or pedestrian, passing a corner moves in a curve, and suffers a centrifugal force, which increases with the velocity, and which impresses on the body a force directed from the corner. An animal causes its weight to resist this force, by voluntarily inclining its body towards the corner. In this case, let A B, _fig. 30._, be the body; C D is the direction of the weight perpendicular to the ground, and C F is the direction of the centrifugal force parallel to the ground and _from_ the corner. The body A B is inclined to the corner, so that the diagonal force (74), which is mechanically equivalent to the weight and centrifugal force, shall be in the direction C A, and shall therefore produce the pressure of the feet upon the ground.

As the velocity is increased, the centrifugal force is also increased, and therefore a greater inclination of the body is necessary to resist it. We accordingly find that the more rapidly a corner is turned, the more the animal inclines his body towards it.

A carriage, however, not having voluntary motion, cannot make this compensation for the disturbing force which is called into existence by the gradual change of direction of the motion; consequently it will, under certain circumstances, be overturned, falling of course outwards, or _from_ the corner. If A B be the carriage, and C, _fig. 31._, the place at which the weight is principally collected, this point C will be under the influence of two forces: the weight, which may be represented by the perpendicular C D, and the centrifugal force, which will be represented by a line C F, which shall have the same proportion to C D as the centrifugal force has to the weight. Now the combined effect of these two forces will be the same as the effect of a single force, represented by C G. Thus, the pressure of the carriage on the road is brought nearer to the outer wheel B. If the centrifugal force bear the same proportion to the weight as C F (or D B), _fig. 32._, bears to C D, the whole pressure is thrown upon the wheel B.

If the centrifugal force bear to the weight a greater proportion than D B has to C D, then the line C F, which represents it, _fig. 33._, will be greater than D B. The diagonal C G, which represents the combined effects of the weight and centrifugal force, will in this case pass outside the wheel B, and therefore this resultant will be unresisted. To perceive how far it will tend to overturn the carriage, let the force C G be resolved into two, one in the direction of C B, and the other C K, perpendicular to C B. The former C B will be resisted by the road, but the latter C K will tend to lift the carriage over the external wheel. If the velocity and the curvature of the course be continued for a sufficient time to enable this force C K to elevate the weight, so that the line of direction shall fall on B, the carriage will be overthrown.

It is evident from what has been now stated, that the chances of overthrow under these circumstances depend on the proportion of B D to C D, or what is to the same purpose, of the distance between the wheels to the height of the principal seat of the load. It will be shown in the next chapter, that there is a certain point, called the centre of gravity, at which the entire weight of the vehicle and its load may be conceived to be concentrated. This is the point which in the present investigation we have marked C. The security of the carriage, therefore, depends on the greatness of the distance between the wheels and the smallness of the elevation of the centre of gravity above the road; for either or both of these circumstances will increase the proportion of B D to C D.

(143.) In the equestrian feat exhibited in the ring at the amphitheatre, when the horse moves round with the performer standing on the saddle, both the horse and rider incline continually towards the centre of the ring, and the inclination increases with the velocity of the motion: by this inclination their weights counteract the effect of the centrifugal force, exactly as in the case already mentioned (142.)

_London, Pubd. by Longman & Co._]

(144.) If a body be allowed to fall by its weight down a convex surface, such as A B, _fig. 34._, it would continue upon the surface until it arrive at B but for the effect of the centrifugal force: this, giving it a motion from the centre of the curve, will cause it to quit the curve at a certain point C, which can be easily found by mathematical computation.

(145.) The most remarkable and important manifestation of centrifugal force is observed in the effects produced by the rotation of the earth upon its axis. Let the circle in _fig. 35._ represent a section of the earth, A B being the axis on which it revolves. This rotation causes the matter which composes the mass of the earth to revolve in circles round the different points of the axis as centres at the various distances at which the component parts of this mass are placed. As they all revolve with the same angular velocity, they will be affected by centrifugal forces, which will be greater or less in proportion as their distances from the centre are greater or less. Consequently the parts of the earth which are situated about the equator, D, will be more strongly affected by centrifugal force than those about the poles, A B. The effect of this difference has been that the component matter about the equator has actually been driven farther from the centre than that about the poles, so that the figure of the earth has swelled out at the sides, and appears proportionally depressed at the top and bottom, resembling the shape of an orange. An exaggerated representation of this figure is given in _fig. 36._; the real difference between the distances of the poles and equator from the centre being too small to be perceptible in a diagram. The exact proportion of C A to C D has never yet been certainly ascertained. Some observations make C D exceed C A by 1/277, and others by only 1/333. The latter, however, seems the more probable. It may be considered to be included between these limits.

The same cause operates more powerfully in other planets which revolve more rapidly on their axes. Jupiter and Saturn have forms which are considerably more elliptical.

(146.) The centrifugal force of the earth’s rotation also affects detached bodies on its surface. If such bodies were not held upon the surface by the earth’s attraction, they would be immediately flung off by the whirling motion in which they participate. The centrifugal force, however, really diminishes the effects of the earth’s attraction on those bodies, or, what is the same, diminishes their weights. If the earth did not revolve on its axis, the weight of bodies in all places equally distant from the centre would be the same; but this is not so when the bodies, as they do, move round with the earth. They acquire from the centrifugal force a tendency to fly from the axis, which increases with their distance from that axis, and is therefore greater the nearer they are to the equator, and less as they approach the pole. But there is another reason why the centrifugal force is more efficient, in the opposition which it gives to gravity near the equator than near the poles. This force does not act from the centre of the earth, but is directed from the earth’s axis. It is, therefore, not directly opposed to gravity, except on the equator itself. On leaving the equator, and proceeding towards the poles, it is less and less opposed to gravity, as will be plain on inspecting _fig. 35._, where the lines P C all represent the direction of gravity, and the lines P F represent the direction of the centrifugal force.

Since, then, as we proceed from the equator towards the poles, not only the amount of the centrifugal force is continually diminished, but also it acts less and less in opposition to gravity, it follows that the weights of bodies are most diminished by it at the equator, and less so towards the poles.

Since bodies are commonly weighed by balancing them against other bodies of known weight, it may be asked, how the phenomena we have been just describing can be ascertained as a matter of fact? for whatever be the body against which it may be balanced, that body must suffer just as much diminution of weight as every other, and consequently, all being diminished in the same proportion, the balance will be preserved though the weights be changed.

To render this effect observable, it will be necessary to compare the effects of gravity with some phenomenon which is not affected by the centrifugal force of the earth’s rotation, and which will be the same at every part of the earth. The means of accomplishing this will be explained in a subsequent chapter.

CHAP. IX.

THE CENTRE OF GRAVITY.

(147.) By the earth’s attraction, all the particles which compose the mass of a body are solicited by equal forces in parallel directions downwards. If these component particles were placed in mere juxtaposition, without any mechanical connection, the force impressed on any one of them could in nowise affect the others, and the mass would, in such a case, be contemplated as an aggregation of small particles of matter, each urged by an independent force. But the bodies which are the subjects of investigation in mechanical science are not found in this state. Solid bodies are coherent masses, the particles of which are firmly bound together, so that any force which affects one, being modified according to circumstances, will be transmitted through the whole body. Liquids accommodate themselves to the shape of the surfaces on which they rest, and forces affecting any one part are transmitted to others, in a manner depending on the peculiar properties of this class of bodies.

As all bodies, which are subjects of mechanical enquiry, on the surface of the earth, must be continually influenced by terrestrial gravity, it is desirable to obtain some easy and summary method of estimating the effect of this force. To consider it, as is unavoidable in the first instance, the combined action of an infinite number of equal and parallel forces soliciting the elementary molecules downwards, would be attended with manifest inconvenience. An infinite number of forces, and an infinite subdivision of the mass, would form parts of every mechanical problem.

To overcome this difficulty, and to obtain all the ease and simplicity which can be desired in elementary investigations, it is only necessary to determine some force, whose single effect shall be equivalent to the combined effects of the gravitation of all the molecules of the body. If this can be accomplished, that single force might be introduced into all problems to represent the whole effect of the earth’s attraction, and no regard need be had to any particles of the body, except that on which this force acts.

(148.) To discover such a force, if it exist, we shall first enquire what properties must necessarily characterise it. Let A B, _fig. 37._, be a solid body placed near the surface of the earth. Its particles are all solicited downwards, in the directions represented by the arrows. Now, if there be any single force equivalent to these combined effects, two properties may be at once assigned to it: 1. It must be presented downwards, in the common direction of those forces to which it is mechanically equivalent; and, 2. it must be equal in intensity to their sum, or, what is the same, to the force with which the whole mass would descend. We shall then suppose it to have this intensity, and to have the direction of the arrow D E. Now, if the single force, in the direction D E, be equivalent to all the separate attractions which affect the particles, we may suppose all these attractions removed, and the body A B influenced only by a single attraction, acting in the direction D E. This being admitted, it follows that if the body be placed upon a prop, immediately under the direction of the line D E, or be suspended from a fixed point immediately above its direction, it will remain motionless. For the whole attracting force in the direction D E will, in the one case, press the body on the prop, and, in the other case, will give tension to the cord, rod, or whatever other means of suspension be used.

(149.) But suppose the body were suspended from some point P, not in the direction of the line D E. Let P C be the direction of the thread by which the body is suspended. Its whole weight, according to the supposition which we have adopted, must then act in the direction C E. Taking C F to represent the weight; it may be considered as mechanically equivalent to two forces (74), C I and C H. Of these C H, acting directly from the point P, merely produces pressure upon it, and gives tension to the cord P C; but C I, acting at right angles to C P, produces motion round P as a centre, and in the direction C I, towards a vertical line P G, drawn through the point P. If the body A B had been on the other side of the line P G, it would have moved in like manner towards it, and therefore in the direction contrary to its present motion.

Hence we must infer, that when the body is suspended from a fixed point, it cannot remain at rest, if that fixed point be not placed in the direction of the line D E; and, on the other hand, that if the fixed point _be_ in the direction of that line, it cannot move. A practical test is thus suggested, by which the line D E may be at once discovered. Let a thread be attached to any point of the body, and let it be suspended by this thread from a hook or other fixed point. The direction of the thread, when the body becomes quiescent, will be that of a single force equivalent to the gravitation of all the component parts of the mass.

(150.) An enquiry is here suggested: does the direction of the equivalent force thus determined depend on the position of the body with respect to the surface of the earth, and how is the direction of the equivalent force affected by a change in that position? This question may be at once solved if the body be suspended by different points, and the directions which the suspending thread takes in each case relatively to the figure and dimensions of the body examined.

The body being suspended in this manner from any point, let a small hole be bored through it, in the exact direction of the thread, so that if the thread were continued below the point where it is attached to the body, it would pass through this hole. The body being successively suspended by several different points on its surface, let as many small holes be bored through it in the same manner. If the body be then cut through, so as to discover the directions which the several holes have taken, they will be all found to cross each other at one point within the body; or the same fact may be discovered thus: a thin wire, which nearly fills the holes being passed through any one of them, it will be found to intercept the passage of a similar wire through any other.

This singular fact teaches us, what indeed can be proved by mathematical reasoning without experiment, that there is _one_ point in every body through which the single force, which is equivalent to the gravitation of all its particles, must pass, in whatever position the body be placed. This point is called _the centre of gravity_.

(151.) In whatever situation a body may be placed, the centre of gravity will have a tendency to descend in the direction of a line perpendicular to the horizon, and which is called the _line of direction_ of the weight. If the body be altogether free and unrestricted by any resistance or impediment, the centre of gravity will actually descend in this direction, and all the other points of the body will move with the same velocity in parallel directions, so that during its fall the position of the parts of the body, with respect to the ground, will be unaltered. But if the body, as is most usual, be subject to some resistance or restraint, it will either remain unmoved, its weight being expended in exciting pressure on the restraining points or surfaces, or it will move in a direction and with a velocity depending on the circumstances which restrain it.

In order to determine these effects, to predict the pressure produced by the weight if the body be quiescent, or the mixed effects of motion and pressure, if it be not so, it is necessary in all cases to be able to assign the place of the centre of gravity. When the magnitude and figure of the body, and the density of the matter which occupies its dimensions, are known, the place of the centre of gravity can be determined with the greatest precision by mathematical calculation. The process by which this is accomplished, however, is not of a sufficiently elementary nature to be properly introduced into this treatise. To render it intelligible would require the aid of some of the most advanced analytical principles; and even to express the position of the point in question, except in very particular instances, would be impossible, without the aid of peculiar symbols.

(152.) There are certain particular forms of body in which, when they are uniformly dense, the place of the centre of gravity can be easily assigned, and proved by reasoning, which is generally intelligible; but in all cases whatever, this point may be easily determined by experiment.

(153.) If a body uniformly dense have such a shape that a point may be found on either side of which in all directions around it the materials of the body are similarly distributed, that point will obviously be the centre of gravity. For if it be supported, the gravitation of the particles on one side drawing them downwards, is resisted by an effect of exactly the same kind and of equal amount on the opposite side, and so the body remains balanced on the point.

The most remarkable body of this kind is a globe, the centre of which is evidently its centre of gravity.

A figure, such as _fig. 38._, called an _oblate spheroid_, has its centre of gravity at its centre, C. Such is the figure of the earth. The same may be observed of the elliptical solid, _fig. 39._, which is called a prolate spheroid.

A cube, and some other regular solids, bounded by plane surfaces, have a point within them, such as above described, and which is, therefore, their centre of gravity. Such are _fig. 40._

A straight wand of uniform thickness has its centre of gravity at the centre of its length; and a cylindrical body has its centre of gravity in its centre, at the middle of its length or axis. Such is the point C, _fig. 41._

A flat plate of any uniform substance, and which has in every part an equal thickness, has its centre of gravity at the middle of its thickness, and under a point of its surface, which is to be determined by its shape. If it be circular or elliptical, this point is its centre. If it have any regular form, bounded by straight edges, it is that point which is equally distant from its several angles, as C in _fig. 42._

(154.) There are some cases in which, although the place of the centre of gravity is not so obvious as in the examples just given, still it may be discovered without any mathematical process, which is not easily understood. Suppose A B C, _fig. 43._, to be a flat triangular plate of uniform thickness and density. Let it be imagined to be divided into narrow bars, by lines parallel to the side A C, as represented in the figure. Draw B D from the angle B to the middle point D of the side A C. It is not difficult to perceive, that B D will divide equally all the bars into which the triangle is conceived to be divided. Now if the flat triangular plate A B C be placed in a horizontal position on a straight edge coinciding with the line B D, it will be balanced: for the bars parallel to A C will be severally balanced by the edge immediately under their middle point; since that middle point is the centre of gravity of each bar. Since, then, the triangle is balanced on the edge, the centre of gravity must be somewhere immediately over it, and must, therefore, be within the plate at some point under the line B D.

The same reasoning will prove that the centre of gravity of the plate is under the line A E, drawn from the angle A to the middle point E of the side B C. To perceive this, it is only necessary to consider the triangle divided into bars parallel to B C, and thence to show that it will be balanced on an edge placed under A E. Since then the centre of gravity of the plate is under the line B D, and also under A E, it must be under the point G, at which these lines cross each other; and it is accordingly at a depth beneath G, equal to half the thickness of the plate.

This may be experimentally verified by taking a piece of tin or card, and cutting it into a triangular form. The point G being found by drawing B D and A E, which divide two sides equally, it will be balanced if placed upon the point of a pin at G.

The centre of gravity of a triangle being thus determined, we shall be able to find the position of the centre of gravity of any plate of uniform thickness and density which is bounded by straight edges, as will be shown hereafter. (173.)

(155.) The centre of gravity is not always included within the volume of the body, that is, it is not enclosed by its surfaces. Numerous examples of this can be produced. If a piece of wire be bent into any form, the centre of gravity will rarely be in the wire. Suppose it be brought to the form of a ring. In that case, the centre of gravity of the wire will be the centre of the circle, a point not forming any part of the wire itself: nevertheless this point may be proved to have the characteristic property of the centre of gravity; for if the ring be suspended by any point, the centre of the ring must always settle itself under the point of suspension. If this centre could be supposed to be connected with the ring by very fine threads, whose weight would be insignificant, and which might be united by a knot or otherwise at the centre, the ring would be balanced upon a point placed under the knot.

In like manner, if the wire be formed into an ellipse, or any other curve similarly arranged round a centre point, that point will be its centre of gravity.

(156.) To find the centre of gravity experimentally, the method described in (149, 150) may be used. In this case two points of suspension will be sufficient to determine it; for the directions of the suspending cord being continued through the body, will cross each other at the centre of gravity. These directions may also be found by placing the body on a sharp point, and adjusting it so as to be balanced upon it. In this case a line drawn through the body directly upwards from the point will pass through the centre of gravity, and therefore two such lines must cross at that point.

(157.) If the body have two flat parallel surfaces like sheet metal, stiff paper, card, board, &c., the centre of gravity may be found by balancing the body in two positions on an horizontal straight edge. The point where the lines marked by the edge cross each other will be immediately under the centre of gravity. This may be verified by showing that the body will be balanced on a point thus placed, or that if it be suspended, the point thus determined will always come under the point of suspension.

The position of the centre of gravity of such bodies may also be found by placing the body on an horizontal table having a straight edge. The body being moved beyond the edge until it is in that position in which the slightest disturbance will cause it to fall, the centre of gravity will then be immediately over the edge. This being done in two positions, the centre of gravity will be determined as before.

(158.) It has been already stated, that when the body is perfectly free, the centre of gravity must necessarily move downwards, in a direction perpendicular to an horizontal plane. When the body is not free, the circumstances which restrain it generally permit the centre of gravity to move in certain directions, but obstruct its motion in others. Thus if a body be suspended from a fixed point by a flexible cord, the centre of gravity is free to move in every direction except those which would carry it farther from the point of suspension than the length of the cord. Hence if we conceive a globe or sphere to surround the point of suspension on every side to a distance equal to that of the centre of gravity from the point of suspension, when the cord is fully stretched, the centre of gravity will be at liberty to move in every direction within this sphere.

There are an infinite variety of circumstances under which the motion of a body may be restrained, and in which a most important and useful class of mechanical problems originate. Before we notice others, we shall, however, examine that which has just been described more particularly.

Let P, _fig. 44._, be the point of suspension, and C the centre of gravity, and suppose the body so placed that C shall be within the sphere already described. The cord will therefore be slackened, and in this state the body will be free. The centre of gravity will therefore descend in the perpendicular direction until the cord becomes fully extended; the tension will then prevent its further motion in the perpendicular direction. The downward force must now be considered as the diagonal of a parallelogram, and equivalent to two forces C D and C E, in the directions of the sides, as already explained in (149). The force C D will bring the centre of gravity into the direction P F, perpendicularly under the point of suspension. Since the force of gravity acts continually on C in its approach to P F, it will move towards that line with accelerated speed, and when it has arrived there it will have acquired a force to which no obstruction is immediately opposed, and consequently by its inertia it retains this force, and moves beyond P F on the other side. But when the point C gets into the line P F, it is in the lowest possible position; for it is at the lowest point of the sphere which limits its motion. When it passes to the other side of P F, it must therefore begin to ascend, and the force of gravity, which, in the former case, accelerated its descent, will now for the same reason, and with equal energy, oppose its ascent. This will be easily understood. Let C′ be any point which it may have attained in ascending; C′ G′, the force of gravity, is now equivalent to C′ D′ and C′ E′. The latter as before produces tension; but the former C′ D′ is in a direction immediately opposed to the motion, and therefore retards it. This retardation will continue until all the motion acquired by the body in its descent from the first position has been destroyed, and then it will begin to return to P F, and so it will continue to vibrate from the one side to the other until the friction on the point P, and the resistance of the air, gradually deprive it of its motion, and bring it to a state of rest in the direction P F.

But for the effects of friction and atmospheric resistance, the body would continue for ever to oscillate equally from side to side of the line P F.

(159.) The phenomenon just developed is only an example of an extensive class. Whenever the circumstances which restrain the body are of such a nature that the centre of gravity is prevented from descending below a certain level, but not, on the other hand, restrained from rising above it, the body will remain at rest if the centre of gravity be placed at the lowest limit of its level; any disturbance will cause it to oscillate around this state, and it cannot return to a state of rest until friction or some other cause have deprived it of the motion communicated by the disturbing force.

(160.) Under the circumstances which we have just described, the body could not maintain itself in a state of rest in any position except that in which the centre of gravity is, at the lowest point of the space in which it is free to move. This, however, is not always the case. Suppose it were suspended by an inflexible rod instead of a flexible string; the centre of gravity would then not only be prevented from receding from the point of suspension, but also from approaching it; in fact, it would be always kept at the same distance from it. Thus, instead of being capable of moving anywhere within the sphere, it is now capable of moving on its surface only. The reasoning used in the last case may also be applied here, to prove that when the centre of gravity is on either side of the perpendicular P F, it will fall towards P F and oscillate, and that if it be placed in the line P F, it will remain in equilibrium. But in this case there is another position, in which the centre of gravity may be placed so as to produce equilibrium. If it be placed at the highest point of the sphere in which it moves, the whole force acting on it will then be directed on the point of suspension, perpendicularly downwards, and will be entirely expended in producing pressure on that point; consequently, the body will in this case be in equilibrium. But this state of equilibrium is of a character very different from that in which the centre of gravity was at the lowest part of the sphere. In the present case any displacement, however slight, of the centre of gravity, will carry it to a lower level, and the force of gravity will then prevent its return to its former state, and will impel it downwards until it attain the lowest point of the sphere, and round that point it will oscillate.

(161.) The two states of equilibrium which have been just noticed, are called stable and instable equilibrium. The character of the former is, that any disturbance of the state produces oscillation about it; but any disturbance of the latter state produces a total overthrow, and finally causes oscillation around the state of stable equilibrium.

Let A B, _fig. 45._, be an elliptical board resting on its edge on an horizontal plane. In the position here represented, the extremity P of the lesser axis being the point of support, the board is in stable equilibrium; for any motion on either side must cause the centre of gravity C to ascend in the directions C O, and oscillation will ensue. If, however, it rest upon the smaller end, as in _fig. 46._, the position would still be a state of equilibrium, because the centre of gravity is directly above the point of support; but it would be instable equilibrium, because the slightest displacement of the centre of gravity would cause it to descend.

Thus an egg or a lemon may be balanced on the end, but the least disturbance will overthrow it. On the contrary, it will easily rest on the side, and any disturbance will produce oscillation.

(162.) When the circumstances under which the body is placed allow the centre of gravity to move only in an horizontal line, the body is in a state which may be called _neutral equilibrium_. The slightest force will move the centre of gravity, but will neither produce oscillation nor overthrow the body, as in the last two cases.

An example of this state is furnished by a cylinder placed upon an horizontal plane. As the cylinder is rolled upon the plane, the centre of gravity C, _fig. 47._, moves in a line parallel to the plane A B, and distant from it by the radius of the cylinder. The body will thus rest indifferently in any position, because the line of direction always falls upon a point P at which the body rests upon the plane.

If the plane were inclined, as in _fig. 48._, a body might be so shaped, that while it would roll the centre of gravity would move horizontally. In this case the body would rest indifferently on any part of the plane, as if it were horizontal, provided the friction be sufficient to prevent the body from sliding down the plane.

If the centre of gravity of a cylinder happen not to coincide with its centre by reason of the want of uniformity in the materials of which it is composed, it will not be in a state of neutral equilibrium on an horizontal plane, as in _fig. 47._ In this case let G, _fig. 49._, be the centre of gravity. In the position here represented, where the centre of gravity is immediately _below_ the centre C, the state will be stable equilibrium, because a motion on either side would cause the centre of gravity to ascend; but in _fig. 50._, where G is immediately above C, the state is instable equilibrium, because a motion on either side would cause G to descend, and the body would turn into the position _fig. 49._

(163.) A cylinder of this kind will, under certain circumstances, roll up an inclined plane. Let A B, _fig. 51._, be the inclined plane, and let the cylinder be so placed that the line of direction from G shall be _above_ the point P at which the cylinder rests upon the plane. The whole weight of the body acting in the direction G D will obviously cause the cylinder to roll towards A, provided the friction be sufficient to prevent sliding; but although the cylinder in this case ascends, the centre of gravity G really descends.

When G is so placed that the line of direction G D shall fall on the point P, the cylinder will be in equilibrium, because its weight acts upon the point on which it rests. There are two cases represented in _fig. 52._ and _fig. 53._, in which G takes this position. _Fig. 52._ represents the state of stable, and _fig. 53._ of instable equilibrium.

(164.) When a body is placed upon a base, its stability depends upon the position of the line of direction and the height of the centre of gravity above the base. If the line of direction fall within the base, the body will stand firm; if it fall on the edge of the base, it will be in a state in which the slightest force will overthrow it on that side at which the line of direction falls; and if the line of direction fall without the base, the body must turn over that edge which is nearest to the line of direction.