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.

Fig. 74.

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].

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].

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.

Fig. 77.

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].

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]).