(128.) In the last chapter, we investigated the phenomena of bodies descending freely in the vertical direction, and determined the laws which govern, not their motion alone, but that of bodies urged by any uniformly accelerating force whatever. We shall now consider some of the most ordinary cases in which the free descent of bodies is impeded, and the effects of their gravitation modified.

(129.) If a body, urged by any forces whatever, be placed upon a hard unyielding surface, it will evidently remain at rest, if the resultant (76) of all the forces which are applied to it be directed perpendicularly against the surface. In this case, the effect produced is pressure, but no motion ensues. If only one force act upon the body, it will remain at rest, provided the direction of that force be perpendicular to the surface.

But the effect will be different, if the resultant of the forces which are applied to the body be oblique to the surface. In that case this resultant, which, for simplicity, may be taken as a single force, may be considered as mechanically equivalent to two forces (76), one in the direction of the surface, and the other perpendicular to it. The latter element will be resisted, and will produce a pressure; the former will cause the body to move. This will perhaps be more clearly apprehended by the aid of a diagram.

Let A B, fig. 25., be the surface, and let P be a particle of matter placed upon it, and urged by a force in the direction P D, perpendicular to A B. It is manifest, that this force can only press the particle P against A B, but cannot give it any motion.

But let us suppose, that the force which urges P is in a direction P F, oblique to A B. Taking P F as the diagonal of a parallelogram, whose sides are P D and P C (74), the force P F is mechanically equivalent to two forces, expressed by the lines P D and P C. But P D, being perpendicular to A B, produces pressure without motion, and P C, being in the direction of A B, produces motion without pressure. Thus the effect of the force P F is distributed between motion and pressure in a certain proportion, which depends on the obliquity of its direction to that of the surface. The two extreme cases are, 1. When it is in the direction of the surface; it then produces motion without pressure: and, 2. When it is perpendicular to the surface; it then produces pressure without motion. In all intermediate directions, however, it will produce both these effects.

(130.) It will be very apparent, that the more oblique the direction of the force P F is to A B, the greater will be that part of it which produces motion, and the less will be that which produces pressure. This will be evident by inspecting fig. 26. In this figure the line P F, which represents the force, is equal to P F in fig. 25. But P D, which expresses the pressure, is less in fig. 26. than in fig. 25., while P C, which expresses the motion, is greater. So long, then, as the obliquity of the directions of the surface and the force remain unchanged, so long will the distribution of the force between motion and pressure remain the same; and therefore, if the force itself remain the same, the parts of it which produce motion and pressure will be respectively equal.

(131.) These general principles being understood, no difficulty can arise in applying them to the motion of bodies urged on inclined planes or curves by the force of gravity. If a body be placed on an unyielding horizontal plane, it will remain at rest, producing a pressure on the plane equal to the total amount of its weight. For in this case the force which urges the body, being that of terrestrial gravity, its direction is vertical, and therefore perpendicular to the horizontal plane.

But if the body P, fig. 25., be placed upon a plane A B, oblique to the direction of the force of gravity, then, according to what has been proved (129), the weight of the body will be distributed into two parts, P C and P D; one, P D, producing a pressure on the plane A B, and the other, P C, producing motion down the plane. Since the obliquity of the perpendicular direction P F of the weight to that of the plane A B must be the same on whatever part of the plane the weight may be placed, it follows (130), that the proportion P C of the weight which urges the body down the plane must be the same throughout its whole descent.

(132.) Hence it may easily be inferred, that the force down the plane is uniform; for since the weight of the body P is always the same, and since its proportion to that part which urges it down the plane is the same, it follows that the quantity of this part cannot vary. The motion of a heavy body down an inclined plane is therefore an uniformly-accelerated motion, and is characterised by all the properties of uniformly-accelerated motion, explained in the last chapter.

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/T²;

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

C = c × W × R/T².

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/T².

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 × N².

(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.)

(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.