THE NATURE OF CIRCULAR MOTION.

557. To compel a body to swerve from motion in a straight line force must be exercised. In this chapter we shall study the comparatively simple case of a body revolving in a circle.

558. When a body moves round uniformly in a circle force must be continuously applied, and the first question for us to examine is, as to the direction of that force. We have to demonstrate the important fact, that it constantly tends towards the centre.

559. The direction of the force can be exhibited by actual experiment, and its magnitude will be at the same time clearly indicated by the extent to which a spring is stretched. The apparatus we use is shown in Fig. 74.

The essential parts of the machine consist of two balls a, b, each 2" in diameter: these are thin hollow spheres of silvered brass. The balls are supported on arms p a, q b, which are attached to a piece of wood, p q, capable of turning on a socket at c. The arm a p is rigidly fixed to p q; the other arm, b q, is capable of turning round a pin at q. An india-rubber door-spring is shown at f; one end of this is secured to p q, the other end to the movable arm, q b. If the arm q b be turned so as to move b away from c, the spring f must be stretched.

A small toothed wheel is mounted on the same socket as c; this is behind p q, and is therefore not seen in the figure: the whole is made to revolve rapidly by the large wheel e, which is turned by the handle d.

560. The room being darkened, a beam from the lime-light is allowed to fall on the apparatus: the reflections of the light are seen in the two silvered balls as two bright points. When d is turned, the balls move round rapidly, and you see the points of light reflected from them describe circles. The ball b when at rest is 4" from c, while a is 8" from c; hence the circle described by b is smaller than that described by a. The appearance presented is that of two concentric luminous circles. As the speed increases, the inner circle enlarges till the two circles blend into one. By increasing the speed still more, you see the circle whose diameter is enlarging actually exceeding the fixed circle, and its size continues to increase until the highest velocity which it is safe to employ has been communicated to the machine.

561. What is the explanation of this? The arm a is fixed and the distance a c cannot alter, hence a describes the fixed circle. b, on the other hand, is not fixed; it can recede from c, and we find that the quicker the speed the further it recedes. The larger the circle described by b the more is the spring stretched, and the greater is the force with which b is attracted towards the centre. This experiment proves that the force necessary to retain a body in a circular path must be increased when the speed is increased.

562. Thus we see that uniform motion of a body in a circle can only be produced by an uniform force directed to the centre.

If the motion, even though circular, have variable speed the law of the force is not so simple.

563. We can measure the magnitude of this force by the same apparatus. The ball b weighs 0·1 lb. I find that I must pull it with a force of 3 lbs. in order to draw it to a distance of 8" from c; that is, to the same distance as a is from c. Hence, when the diameters of the circles in which the balls move are equal, the central force must be 3 lbs.; that is, it must be nearly thirty times as great as gravity.

564. The necessity for the central force is thus shown: Let us conceive a weight attached to a string to be swung round in a circle, a portion of which is shown in Fig. 75.

Suppose the weight be at s and moving towards p, and let a tangent to the circle be drawn at p. Take two points on the circle, a and B, very near p; the small arc a b does not differ perceptibly from the part a b on the tangent line; hence, when the particle arrives at a, it is a matter of indifference whether it travels in the arc a b, or along the line a b. Let us suppose it to move along the line. By the first law of motion, a particle moving in the line a b would continue to do so; hence, if the particle be allowed, it will move on to q: but the particle is not allowed to move to q; it is found at r. Hence it must have been withdrawn by some force.

565. This force is supplied by the string to which the weight is attached. The incessant change from the rectilinear motion of the weight requires a constantly applied force, and this is always directed to the centre. Should the string be released, the body flies off in the direction of the tangent to the circle at the point which the body occupied at the instant of release.

566. The central force increases in proportion to the square of the velocity. If I double the speed with which the weight is whirled round in the circle, I quadruple the force which the string must exert on the body. If the speed be trebled, the force is increased ninefold, and so on. When the speeds with which two equal masses are revolving in two circles are equal, the central force in the smaller circle is greater than that of the larger circle, in the proportion of the radius of the larger circle to that of the smaller.

THE ACTION OF CIRCULAR MOTION
UPON LIQUIDS.

567. I have here a small bucket nearly filled with water: to the handle a piece of string is attached. If I whirl the bucket round in a vertical plane sufficiently fast, you see no water escapes, although the bucket is turned upside down once in every revolution. This is because the water has not time to fall out during such a brief interval. A body would not fall half an inch from rest in the twentieth of a second.

568. The action of circular motion upon liquids is illustrated by the experiment which is represented in Fig. 76.

A glass beaker about half full of water is mounted so that it can be spun round rapidly. The motion is given by means of a large wheel turned by a handle, as shown in the figure. When the rotation commences, the water is seen to rise up against the glass sides and form a hollow in the centre.

569. In order to demonstrate this clearly, I turn upon the vessel a beam from the lime-light. I have previously dissolved a little quinine in the water. The light from the lamp is transmitted through a piece of dense blue glass. When the light thus coloured falls on the water, the presence of the quinine makes the entire liquid glow with a bluish hue. This remarkable property of quinine, which is known as fluorescence, enables you to see distinctly the hollow form caused by the rotation.

570. You observe that as the speed becomes greater the depth of the hollow increases, and that if I turn the wheel sufficiently fast the water is actually driven out of the glass. The shape of the curve which the water assumes is that which would be produced by the revolution of a parabola about its axis.

571. The explanation is simple. As soon as the glass begins to revolve, the friction of its sides speedily imparts a revolving motion to the water; but in this case there is nothing to keep the particles near the centre like the string in the revolving weight, so the liquid rises at the sides of the glass.

572. But you may ask why all the particles of the water should not go to the circumference, and thus line the inside of the glass with a hollow cylinder of water instead of the parabola. Such an arrangement could not exist in a liquid acted on by gravity. The lower parts of the cylinder must bear the pressure of the water above, and therefore have more tendency to flatten out than the upper portions. This tendency could not be overcome by any consequences of the movement, for such must be alike on all parts at the same distance from the axis.

573. A very beautiful experiment was devised by Plateau for the purpose of studying the revolution of a liquid removed from the action of gravity.

The apparatus employed is represented in Fig. 77. A glass vessel 9" cube is filled with a mixture of alcohol and water. The relative quantities ought to be so proportioned that the fluid has the same specific gravity as olive oil, which is heavier than alcohol and lighter than water. In practice, however, it is found so difficult to adjust the composition exactly that the best plan is to make two alcoholic mixtures so that olive oil will just float on one of them, and just sink in the other. The lower half of the glass is to be filled with the denser mixture, and the upper half with the lighter. If, then, the oil be carefully introduced with a funnel it will form a beautiful sphere in the middle of the vessel, as shown in the figure. We thus see that a liquid mass freed from the action of terrestrial gravity, forms a sphere by the mutual attraction of its particles.

Through the liquid a vertical spindle passes. On this there is a small disk at the middle of its length, about which the sphere of oil arranges itself symmetrically. To the end of the spindle a handle is attached. When the handle is turned round slowly, the friction of the disk and spindle communicates a motion of rotation to the sphere of oil. We have thus a liquid spheroidal mass endowed with a movement of rotation; and we can study the effect of the motion upon its form. We first see the sphere flatten down at its poles, and bulge at its equator. In order to show the phenomenon to those who may not be near to the table, the sphere can be projected on the screen by the help of the lime-light lamp and a lens. It first appears as a yellow circle, and then, as the rotation begins, the circle gradually transforms into an ellipse. But a very remarkable modification takes place when the handle is turned somewhat rapidly. The ellipsoid gradually flattens down until, when a certain velocity has been attained, the surface actually becomes indented at the poles, and flies from the axis altogether. Ultimately the liquid assumes the form of a beautiful ring, and the appearance on the screen is shown in Fig. 78.

574. The explanation of the development of the ring involves some additional principles: as the sphere of oil spins round in the liquid, its surface is retarded by friction; so that when the velocity attains a certain amount, the internal portions of the sphere, which are in the neighbourhood of the spindle, are driven from the centre into the outer portions, but the full account of the phenomenon cannot be given here.

575. The earth was, we believe, originally in a fluid condition. It had then, as it has now, a diurnal rotation, and one of the consequences of this rotation has been to cause the form to be slightly protuberant at the equator, just as we have seen the sphere of oil to bulge out under similar conditions.

576. Bodies lying on the earth are whirled around in a great circle every day. Hence, if there were not some force drawing them to the centre, they would fly off at a tangent. A part of the earth’s attraction goes for this purpose, and the remainder, which is the apparent weight, is thus diminished by a quantity increasing from the pole to the equator (Art. 86).

THE APPLICATIONS OF CIRCULAR MOTION.

577. These principles have many applications in the mechanical arts; we shall mention two of them. The first is to the governor-balls of a steam-engine; the second is to the process of sugar refining.

An engine which turns a number of machines in a factory should work uniformly. Irregularities of motion may be productive of loss and various inconveniences. An engine would work irregularly either from variation in the production of steam, or from the demands upon the power being lessened or increased. Even if the first of these sources of irregularity could be avoided by care, it is clear that the second could not. Some machines in the mill are occasionally stopped, others occasionally set in motion, and the engine generally tends to go faster the less it has to do. It is therefore necessary to provide means by which the speed shall be restrained within narrow limits, and it is obviously desirable that the contrivance used for this purpose should be self-acting. We must, therefore, have some arrangement which shall admit more steam to the cylinder when the engine is moving too slowly, and less steam when it is moving too quickly. The valve which is to regulate this must be worked by some agent which depends upon the velocity of the engine; this at once points to circular motion because the force acting on the revolving body depends upon its velocity. Such was the train of reasoning which led to the happy invention of the governor-balls: these are shown in Fig. 79.

a b is a vertical spindle which is turned by the engine. p p is a piece firmly attached to the spindle and turning with it. p w, p w are arms terminating in weights w w; these are balls of iron, generally very massive: the arms are free to turn round pins at p p. At q q links are placed, attached to another piece r r, which is free to slide up and down the spindle. When a b rotates, w and w are carried round, and therefore fly outwards from the spindle; to do this they must evidently pull the piece r r up the shaft. We can easily imagine an arrangement by which r r shall be made to shut or open the steam-valve according as it ascends or descends. The problem is then solved, for if the engine begin to go too rapidly, the balls fly out further just as they did in Fig. 74: this movement raises the piece r r, which diminishes the supply of steam, and consequently checks the speed. On the other hand, when the engine works too slowly, the balls fall in towards the spindle, the piece r r descends, the valve is opened, and a greater supply of steam is admitted. The objection to this governor is that though it moderates, it does not completely check irregularity. There are other governors occasionally employed which depend also on circular motion; some of these are more sensitive than the governor-balls; but they are elaborate machines, only to be employed under exceptional circumstances.

578. The application of circular motion to sugar refining is a very beautiful invention. To explain it I must briefly describe the process of refining.

The raw sugar is dissolved in water, and the solution is purified by straining and by filtration through animal charcoal. The syrup is then boiled. In order to preserve the colour of the sugar, and to prevent loss, this boiling is conducted in vacuo, as by this means the temperature required is much less than would be necessary with the ordinary atmospheric pressure.

The evaporation having been completed, crystals of sugar form throughout the mass of syrup. To separate these crystals from the liquor which surrounds them, the aid of circular motion force is called in. A mass of the mixture is placed in a large iron tub, the sides of which are perforated with small holes. The tub is then made to rotate with prodigious velocity; its contents instantly fly off to the circumference, the liquid portions find an exit through the perforations in the sides, but the crystals are left behind. A little clear syrup is then sprinkled over the sugar while still rotating: this washes from the crystals the last traces of the coloured liquid, and passes out through the holes; when the motion ceases, the inside of the tub contains a layer of perfectly pure white sugar, some inches in thickness, ready for the market.

579. Circular motion is peculiarly fitted for this purpose; each particle of liquid strives to get as far away from the axis as possible. The action on the sugar is very different from what it would have been had the mass been subjected to pressure by a screw-press or similar contrivance; the particles immediately acted on would then have to transmit the pressure to those within; and the consequence would be that while the crystals of sugar on the outside would be crushed and destroyed, the water would only be very imperfectly driven from the interior: for it could lurk in the interstices of the sugar, which remain notwithstanding the pressure.

580. But with circular motion the water must go, not because it is pushed by the crystals, but because of its own inertia; and it can be perfectly expelled by a velocity of rotation less than that which would be necessary to produce sufficient pressure to make the crystals injure each other.

THE PERMANENT AXES.

581. There are some curious properties of circular motion which remain to be considered. These we shall investigate by means of the apparatus of Fig. 80. This consists of a pair of wheels b c, by which a considerable velocity can be given to a horizontal shaft. This shaft is connected by a pair of bevelled wheels d with a vertical spindle f. The machine is worked by a handle a, and the object to be experimented upon is suspended from the spindle.

582. I first take a disk of wood 18" in diameter; a hole is bored in the margin of this disk; through this hole a rope is fastened, by means of which the disk is suspended from the spindle. The disk hangs of course in a vertical plane.

583. I now begin to turn the handle round gently, and you see the disk begins to rotate about the vertical diameter; but, as the speed increases, the motion becomes a little unsteady; and finally, when I turn the handle very rapidly, the disk springs up into a horizontal plane, and you see it like the surface of a small table: the rope swings round and round in a cone, so rapidly that it is hardly seen.

584. We may repeat the experiment in a different manner. I take a piece of iron chain about 2' long, g; I pass the rope through the two last links of its extremities, and suspend the rope from the spindle. When I commence to turn the handle, you see the chain gradually opens out into a loop h; and as the speed increases, the loop becomes a complete ring. Still increasing the speed, I find the ring becomes unsteady, till finally it rises into a horizontal plane. The ring of chain in the horizontal plane is shown at i. When the motion is further increased, the ring swings about violently, and so I cease turning the handle.

585. The principles already enunciated will explain these remarkable results; we shall only describe that of the chain, as the same explanation will include that of the disk of wood. We shall begin with the chain hanging vertically from the spindle: the moment rotation commences, the chain begins to spin about a vertical axis; the parts of the chain fly outwards from this axis just as the ball flies outwards in Fig. 74; this is the cause of the looped form h which the chain assumes. As the speed is increased the loop gradually opens more and more, just as the diameter of the circle Fig. 74 increases with the velocity. But we have also to inquire into the cause of the remarkable change of position which the ring undergoes; instead of continuing to rotate about a vertical diameter, it comes into a horizontal plane. This will be easily understood with the help of Fig. 81. Let o p represent the rope attached to the ring, and o c be the vertical axis. Suppose the ring to be spinning about the axis o c, when o c was a diameter; if then, from any cause, the ring be slightly displaced, we can show that the circular motion will tend to drive the ring further from the vertical plane, and force it into the horizontal plane. Let the ring be in the position represented in the figure; then, since it revolves about the vertical line o c, the tendency of p p and q q is to move outwards in the directions of the arrows, thus evidently tending to bring the ring into the horizontal plane.

586. In Art. 103 we have explained what is meant by stable and unstable equilibrium; we have here found a precisely analogous phenomenon in motion. The rotation of the ring about its diameter is unstable, for the minutest deviation of the ring from this position is fatal; circumstances immediately combine to augment the deviation more and more, until finally the ring is raised into the horizontal plane. Once in the horizontal plane, the motion there is stable, for if the ring be displaced the tendency is to restore it to the horizontal.

587. The ring, when in a horizontal plane, rotates permanently about the vertical axis through its centre; this axis is called permanent, to distinguish it from all other directions, as being the only axis about which the motion is stable.

588. We may show another experiment with the chain: if instead of passing the rope through the links at its ends, I pass the rope through the centre of the chain, and allow the ends of the chain to hang downwards. I now turn the handle; instantly the parts of the chain fly outwards in a curved form; and by increasing the velocity, the parts of the chain at length come to lie almost in a straight line.