Mathematical Recreations and Essays · W. W. Rouse Ball
I proceed now to enumerate a few questions connected with mechanics which lead to results that seem to me interesting from a historical point of view or paradoxical. Problems in mechanics generally involve more difficulties than problems in arithmetic, algebra, or geometry, and the explanations of some phenomena--such as those connected with the flight of birds--are still incomplete, while the explanations of many others of an interesting character are too difficult to find a place in a non-technical work. Here, however, I shall confine myself to questions which, like those treated in the two preceding chapters, are of an elementary, not to say trivial, character; and the conclusions are well-known to mathematicians.
I assume that the reader is acquainted with the fundamental ideas of kinematics and dynamics, and is familiar with the three Newtonian laws; namely, first that a body will continue in its state of rest or of uniform motion in a straight line unless compelled to change that state by some external force: second, that the change of momentum per unit of time is proportional to the external force and takes place in the direction of it: and third, that the action of one body on another is equal in magnitude but opposite in direction to the reaction of the second body on the first. The first and second laws state the principles required for solving any question on the motion of a particle under the action of given
forces. The third law supplies the additional principle required for the solution of problems in which two or more particles influence one another.
[Paradoxes on Motion][Zeno's Paradoxes.]Motion The difficulties connected with the idea of motion have been for a long time a favourite subject for paradoxes
, some of which bring us into the realm of the philosophy of mathematics.
Zeno's Paradoxes on Motion One of the earliest of these is the remark of Zeno to the effect that since an arrow cannot move where it is not, and since also it cannot move where it is ( in the space it exactly fills), it follows that it cannot move at all. The answer that the very idea of the motion of the arrow implies the passage from where it is to where it is not was rejected by Zeno, who seems to have thought that the appearance of motion of a body was a phenomenon caused by the successive appearances of the body at rest but in different positions.
Zeno also asserted that the idea of motion was itself inconceivable, for what moves must reach the middle of its course before it reaches the end. Hence the assumption of motion presupposes another motion, and that in turn another, and so ad infinitum. His objection was in fact analogous to the biological difficulty expressed by Swift:--
" So naturalists observe, a flea hath smaller fleas that on him prey. And these have smaller fleas to bite 'em. And so proceed ad infinitum."
Or as De Morgan preferred to put it
" Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these have greater still, and greater still, and so on."
Achilles and the Tortoise Zeno's paradox about Achilles and the tortoise is known even more widely. The assertion was that if Achilles ran ten times as fast as a tortoise, yet if the tortoise had (say) 1000 yards start it could never be
overtaken. To establish this, Zeno argued that when Achilles had gone the 1000 yards, the tortoise would still be 100 yards in front of him; by the time he had covered these 100 yards, it would still be 10 yards in front of him; and so on for ever. Thus Achilles would get nearer and nearer to the tortoise but would never overtake it. Zeno regarded this as confirming his view that the popular idea of motion is self-contradictory.
Zeno's Paradox on Time The fallacy of Achilles and the Tortoise is usually explained by saying that though the time required to overtake the tortoise can be divided into an infinite number of intervals, as stated in the argument, yet these intervals get smaller and smaller in geometrical progression, and the sum of them all is a finite time: after the lapse of that time Achilles would be in front of the tortoise. Probably Zeno would have replied that this explanation rests on the assumption that space and time are infinitely divisible, propositions which he would not admit. He seems further to have contended that while, to an accurate thinker, the notion of the infinite divisibility of time was impossible, it was equally impossible to think of a minimum measure of time. For suppose, he argued, that τ is the smallest conceivable interval, and suppose that three horizontal lines composed of three consecutive spans abc , a'b'c' , a"b"c" are placed so that aa'a" , bb'b" , cc'c" are vertically over one another. Imagine the second line moved as a whole one span to the right in the time τ , and simultaneously the third line moved as a whole one span to the left. Then b , a' , c" will be vertically over one another. And in this duration τ (which by hypothesis is indivisible) c' must have passed vertically over a". Hence the duration is divisible, contrary to the hypothesis.
The Paradox of Tristram Shandy. The Paradox of Tristram Shandy Mr Russell has enunciated (B.A.W. Russell, Principles of Mathematics, Cambridge, 1903, vol. i, p. 358.) a paradox somewhat similar to that of Achilles and the Tortoise, save that the intervals of time considered get
longer and longer during the course of events. Tristram Shandy, as we know, took two years writing the history of the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that he could never come to an end, however long he lived. But had he lived long enough, and not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten. For if he wrote the events of the first day in the first year, he would write the events of the n th day in the n th year, hence in time the events of any assigned day would be written, and therefore no part of his biography would remain unwritten. This argument might be put in the form of a demonstration that the part of a magnitude may be equal to the whole of it.
Questions, such as those given above, which are concerned with the continuity and extent of space and time involve difficulties of a high order.
Angular Motion. Angular Motion A non-mathematician finds additional difficulties in the idea of angular motion. For instance, here is a well-known proposition on motion in an equiangular spiral (of which the result is true on the ordinary conventions of mathematics) which shows that a body, moving with uniform velocity and as slowly as we please, may in a finite time whirl round a fixed point an infinite number of times.
The equiangular spiral is the trace of a point P , which moves along a line OP , the line OP turning round a fixed point O with uniform angular velocity while the distance of P from O decreases with the time in geometrical progression. If the radius vector rotates through four right angles we have one convolution of the curve. All convolutions are similar, and the length of each convolution is a constant fraction, say 1/n th, that of the convolution immediately outside it. Inside any given convolution, there are an infinite number of convolutions which get smaller and smaller as we get nearer the pole. Now suppose a point Q to move uniformly along the spiral from any point towards the pole. If it covers the first
convolution in a seconds, it will cover the next in a/n seconds, the next in a/n^2 seconds, and so on, and will finally reach the pole in (a + a/n + a/n^2 + a/n^3 +...) seconds, that is, in an/(n-1) seconds. The velocity is uniform, and yet in a finite time, Q will have traversed an infinite number of convolutions and therefore have circled round the pole an infinite number of times (The proposition is put in this form in J. Richard's Philosophie des math-é-mat-iques, Paris, 1903, pp. 119--120.).
Simple Relative Motion. Simple Relative Motion Even if the philosophical difficulties suggested by Zeno are settled or evaded, the mere idea of relative motion has been often found to present difficulties, and Zeno himself failed to explain a simple phenomenon involving the principle. As one of the easiest examples of this kind, I may quote the common question of how many trains going from B to A a passenger from A to B would meet and pass on his way, assuming that the journey either way takes 41/2 hours and that the trains start from each end every hour. The answer is 9. Or again this: Take two pennies, face upwards on a table and edges in contact. Suppose that one is fixed and that the other rolls on it without slipping, making one complete revolution round it and returning to its initial position. How many revolutions round its own centre has the rolling coin made? The answer is 2.
Laws of Motion Force, Inertia, Centrifugal Force Laws of Motion. I proceed next to make a few remarks on points connected with the laws of motion.
The first law of motion is often said to define force, but it is in only a qualified sense that this is true. Probably the meaning of the law is best expressed in Clifford's phrase, that force is "the description of a certain kind of motion"--in other words it is not an entity but merely a convenient way of stating, without circumlocution, that a certain kind of motion is observed.
It is not difficult to show that any other interpretation lands us in difficulties. Thus some authors use the law to justify a definition that force is that which moves a body or
changes its motion; yet the same writers speak of a steam-engine moving a train. It would seem then that, according to them, a steam-engine is a force. That such statements are current may be fairly reckoned among mechanical paradoxes.
The idea of force is difficult to grasp. How many people, for instance, could predict correctly what would happen in a question as simple as the following? A rope (whose weight may be neglected) hangs over a smooth pulley; it has one end fastened to a weight of 10 stone, and the other end to a sailor of weight 10 stone, the sailor and the weight hanging in the air. The sailor begins to climb up the rope; will the weight move at all; and, if so, will it rise or fall?
It will be noted that in the first law of motion it is asserted that, unless acted on by an external force, a body in motion continues to move (i) with uniform velocity, and (ii) in a straight line.
The tendency of a body to continue in its state of rest or of uniform motion is called its inertia. This tendency may be used to explain various common phenomena and experiments. Thus, if a number of dominoes or draughts are arranged in a vertical pile, a sharp horizontal blow on one of those near the bottom will send it out of the pile, and those above will merely drop down to take its place--in fact they have not time to change their relative positions before there is sufficient space for them to drop vertically as if they were a solid body.
This also is the principle on which depends the successful playing of "Aunt Sally," and the performance of numerous tricks, described in collections of mathematical puzzles (See Les récréations scientifiques by G. Tissandier, where several ingenious illustrations of inertia are given.).
The statement about inertia in the first law may be taken to imply that a body set in rotation about a principal axis passing through its centre of mass will continue to move with a uniform angular velocity and to keep its axis of rotation fixed in direction. The former of these statements is the assumption
on which our measurement of time is based as mentioned below in chapter xiii. The latter assists us to explain the motion of a projectile in a resisting fluid. It affords the explanation of why the barrel of a rifle is grooved; and why, similarly, anyone who has to throw a flat body of irregular shape (such as a card) in a given direction usually gives it a rapid rotatory motion about a principal axis. Elegant illustrations of the fact just mentioned are afforded by a good many of the tricks of acrobats, though the full explanation of most of them also introduces other considerations. Thus when some few years ago the Japanese village at Knightsbridge was one of the shows of London, there were some acrobats there who tossed on to the top surface of an umbrella a penny so that it alighted on its edge, and then, by turning round the stick of the umbrella rapidly, the coin was caused to rotate, but as the umbrella moved away underneath it the coin remained apparently stationary and standing upright, while by diminishing or increasing the angular velocity of the umbrella the penny was caused to run forwards or backwards. This is not a difficult trick to execute.
The tendency of a body in motion to continue to move in a straight line is sometimes called its centrifugal force. Thus, if a train is running round a curve, it tends to move in a straight line, and is constrained only by the pressure of the rails to move in the required direction. Hence it presses on the outer rail of the curve. This pressure can be diminished to some extent both by raising the outer rail, and by putting a guard rail, parallel and close to the inner rail, against which the wheels on that side also will press.
An illustration of this fact occurred in a little known incident of the American civil war (Capturing a Locomotive by W. Pittenger, London, 1882, p. 104.). In the spring of 1862 a party of volunteers from the North made their way to the rear of the Southern armies and seized a train, intending to destroy, as they passed along it, the railway which was the main line of communication between various confederate corps
and their base of operations. They were however detected and pursued. To save themselves, they stopped on a sharp curve and tore up some rails so as to throw the engine which was following them off the line. Unluckily for themselves they were ignorant of dynamics and tore up the inner rails of the curve, an operation which did not incommode their pursuers.
The second law gives us the means of measuring mass, force, and therefore work. A given agent in a given time can do only a definite amount of work. This is illustrated by the fact that although, by means of a rigid lever and a fixed fulcrum, any force however small may be caused to move any mass however large, yet what is gained in power is lost in speed--as the popular phrase runs.
Montucla (Ozanam, 1803 edition, vol. ii, p. 18; 1840 edition, p. 202.) inserted a striking illustration of this principle founded on the well-known story of Archimedes who is said to have declared to Hiero that, were he but given a fixed fulcrum, he could move the world. Montucla calculated the mass of the earth and, assuming that a man could work incessantly at the rate of 116 foot-lbs. per second, which is a very high estimate, he found that it would take over three billion centuries, 3 × 10^14 years, before a mass equal to that of the earth was moved as much as one inch against gravity at the surface of the earth: to move it one inch along a horizonal plane would take about 74000 centuries.
Stability of Equilibrium Work, Stability of Equilibrium, &c. It is known to all those who have read the elements of mechanics that the centre of gravity of a body, which is resting in equilibrium under its own weight, must be vertically above its base: also, speaking generally, we may say that, if every small displacement has the effect of raising the centre of gravity, then the equilibrium is stable, that is, the body when left to itself will return to its original position; but, if a displacement has the effect of lowering the centre of gravity, then for that displacement the equilibrium is unstable; while, if every displacement does not alter the
height above some fixed plane of the centre of gravity, then the equilibrium is neutral. In other words, if in order to cause a displacement work has to be done against the forces acting on the body, then for that displacement the equilibrium is stable, while if the forces do work the equilibrium is unstable.
A good many of the simpler mechanical toys and tricks afford illustrations of this principle.
Magic Bottles. Magic Bottles (Ozanam, 1803 edition, vol. ii, p. 15; 1840 edition, p. 201.) Among the most common of such toys are the small bottles--trays of which may be seen any day in the streets of London--which keep always upright, and cannot be upset until their owner orders them to lie down. Such a bottle is made of thin glass or varnished paper fixed to the plane surface of a solid hemisphere or smaller segment of a sphere. Now the distance of the centre of gravity of a homogeneous hemisphere from the centre of the sphere is three-eighths of the radius, and the mass of the glass or varnished paper is so small compared with the mass of the lead base that the centre of gravity of the whole bottle is still within the hemisphere. Let us denote the centre of the hemisphere by C , and the centre of gravity of the bottle by G.
If such a bottle is placed with the hemisphere resting on a horizontal plane and GC vertical, any small displacement on the plane will tend to raise G , and thus the equilibrium is stable. This may be seen also from the fact that when slightly displaced there is brought into play a couple, of which one force is the reaction of the table passing through C and acting vertically upward, and the other the weight of the bottle acting vertically downward at G. If G is below C , this couple tends to restore the bottle to its original position.
If there is dropped into the bottle a shot or nail so heavy as to raise the centre of gravity of the whole above C , then the equilibrium is unstable, and, if any small displacement is given, the bottle falls over on to its side.
Montucla says that in his time it was not uncommon to see boxes of tin soldiers mounted on lead hemispheres, and
when the lid of the box was taken off the whole regiment sprang to attention.
In a similar way we may explain how to balance a pencil in a vertical position, with its point resting on the top of one's finger, an experiment which is described in nearly every book of puzzles (Oughtred, Mathematical Recreations, p. 24; Ozanam, 1803 edition, vol. ii, p. 14; 1840 edition, p. 200.). This is effected by taking a penknife, of which one blade is opened through an angle of (say) 120^circ , and sticking the blade in the pencil so that the handle of the penknife is below the finger. The centre of gravity is thus brought below the point of support, and a small displacement given to the pencil will raise the centre of gravity of the whole: thus the equilibrium is stable.
Other similar tricks are the suspension of a bucket over the edge of a table by a couple of sticks, and the balancing of a coin on the edge of a wine-glass by the aid of a couple of forks (Oughtred, p. 30; Ozanam, 1803 edition, vol. ii, p. 12; 1840 edition, p. 199.)--the sticks or forks being so placed that the centre of gravity of the whole is vertically below the point of support and its depth below it a maximum.
The toy representing a horseman, whose motion continually brings him over the edge of a table into a position which seems to ensure immediate destruction, is constructed in somewhat the same way. A wire has one end fixed to the feet of the rider; the wire is curved downwards and backwards, and at the other end is fixed a weight. When the horse is placed so that his hind legs are near the edge of the table and his forefeet over the edge, the weight is under his hind feet. Thus the whole toy forms a pendulum with a curved instead of a straight rod. Hence the farther it swings over the table, the higher is the centre of gravity raised, and thus the toy tends to return to its original position of equilibrium.
An elegant modification of the prancing horse was brought out at Paris in 1890 in the shape of a toy made of tin and in
the figure of a man (La Nature, Paris, March, 1891.). The legs are pivoted so as to be movable about the thighs, but with a wire check to prevent too long a step, and the hands are fastened to the top of a -shaped wire weighted at its ends. If the figure is placed on a narrow sloping plank or strip of wood passing between the legs of the , then owing to the -shaped wire any lateral displacement of the figure will raise its centre of gravity, and thus for any such displacement the equilibrium is stable. Hence, if a slight lateral disturbance is given, the figure will oscillate and will rest alternately on each foot: when it is supported by one foot the other foot under its own weight moves forwards, and thus the figure will walk down the plank though with a slight reeling motion. Shortly after the publication of the third edition of this book an improved form of this toy, in the shape of a walking elephant made in heavy metal, was issued in England, and probably in that form it is now familiar to all who are interested in noticing street toys.
Columbus's Egg. Columbus's Egg The toy known as Columbus's egg depends on the same principle as the magic bottle, though it leads to the converse result. The shell of the egg is made of tin and cannot be opened. Inside it and fastened to its base is a hollow truncated tin cone, and there is also a loose marble inside the shell. If the egg is held properly, the marble runs inside the cone and the egg will stand on its base, but so long as the marble is outside the cone, the egg cannot be made to stand on its base.
Cones running up hill. Cones running up hill (Ozanam, 1803 edition, vol. ii, p. 49; 1840 edition, p. 216.) The experiment to make a double cone run up hill depends on the same principle as the toys above described; namely, on the tendency of a body to take a position so that its centre of gravity is as low as possible. In this case it produces the optical effect of a body moving by itself up a hill.
Usually the experiment is performed as follows. Arrange two sticks in the shape of a , with the apex on a table and
the two upper ends resting on the top edge of a book placed on the table. Take two equal cones fixed base to base, and place them with the curved surfaces resting on the sticks near the apex of the , the common axis of the cones being horizontal and parallel to the edge of the book. Then, if properly arranged, the cones will run up the plane formed by the sticks.
The explanation is obvious. The centre of gravity of the cones moves in the vertical plane midway between the two sticks and it occupies a lower position as the points of contact on the sticks get farther apart. Hence as the cone rolls up the sticks its centre of gravity descends
.
The idea of making a machine which
once set going would continue to go for ever by itself has been the ignis fatuus of self-taught mechanicians in much the same way as the quadrature of the circle has been of self-taught geometricians.
Now the obvious meaning of the third law of motion is that a force is only one aspect of a stress, and that whenever a force is caused another equal and opposite one is brought also into existence--though it may act upon a different body, and thus be immaterial for the particular problem considered. The law however is capable of another interpretation (Newton's Principia, last paragraph of the Scholium to the Laws of Motion.), namely, that the rate at which an agent does work (that is, its action) is equal to the rate at which work is done against it (that is, its reaction). If it is allowable to include in the reaction the rate at which kinetic energy is being produced, and if work is taken to include that done against molecular forces, then it follows from this interpretation that the work done by an agent on a system is equivalent to the total increase of energy, that is, the power of doing work. Hence in an isolated system the total amount of energy is constant. If this is granted, then since friction and some molecular dissipation of energy
cannot be wholly prevented, it must be impossible to construct in an isolated system a machine capable of perpetual motion.
I do not propose to describe in detail the various machines for producing perpetual motion which have been suggested, but I may add that a number of them are equivalent essentially to the one of which a section is represented in the accompanying figure.
It consists of two concentric vertical wheels in the same plane, and mounted on a horizontal axle through their centre, C. The space between the wheels is divided into compartments by spokes inclined at a constant angle to the radii to the points whence they are drawn, and each compartment contains a heavy bullet. Apart from these bullets, the wheels would be in equilibrium. Each bullet tends to turn the wheels round their axle, and the moment which measures this tendency is the product of the weight of the bullet and its distance from the vertical through C.
The idea of the constructors of such machines was that, as the bullet in any compartment would roll under gravity to the lowest point of the compartment, the bullets on the right-hand side of the diagram would be farther from the vertical through C than those on the left. Hence the sum of the moments of the weights of the bullets on the right would be greater than the sum of the moments of those on the left. Thus the wheels
would turn continually in the same direction as the hands of a watch. The fallacy in the argument is obvious.
Another large group of machines for producing perpetual motion depended on the use of a magnet to raise a mass which was then allowed to fall under gravity. Thus, if the bob of a simple pendulum was made of iron, it was thought that magnets fixed near the highest points which were reached by the bob in the swing of the pendulum would draw the bob up to the same height in each swing and thus give perpetual motion.
Of course it is only in isolated systems that the total amount of energy is constant, and, if a source of external energy can be obtained from which energy is continually introduced into the system, perpetual motion is, in a sense, possible; though even here materials would ultimately wear out. The solar heat and the tides are among the most obvious of such sources.
There was at Paris in the latter half of the eighteenth century a clock which was an ingenious illustration of such perpetual motion (Ozanam, 1803 edition, vol. ii, p. 105; 1840 edition, p. 238.). The energy which was stored up in it to maintain the motion of the pendulum was provided by the expansion of a silver rod. This expansion was caused by the daily rise of temperature, and by means of a train of levers it wound up the clock. There was a disconnecting apparatus, so that the contraction due to a fall of temperature produced no effect, and there was a similar arrangement to prevent overwinding. I believe that a rise of eight or nine degrees Fahrenheit was sufficient to wind up the clock for twenty-four hours.
I have in my possession a watch, which produces the same effect by somewhat different means. Inside the case is a steel weight, and if the watch is carried in a pocket this weight rises and falls at every step one takes, somewhat after the manner of a pedometer. The weight is raised by the action of the person who has it in his pocket in taking a step, and in falling it winds up the spring of the watch. On the face is a small dial showing the number of hours for
which the watch is wound up. As soon as the hand of this dial points to fifty-six hours, the train of levers which winds up the watch disconnects automatically, so as to prevent overwinding the spring, and it reconnects again as soon as the watch has run down eight hours. The watch is an excellent time-keeper, and a walk of about a couple of miles is sufficient to wind it up for twenty-four hours
.
I may add here the observation, which is well known to mathematicians, but is a perpetual source of disappointment to ignorant inventors, that it frequently happens that an accurate model of a machine will work satisfactorily while the machine itself will not do so.
One reason for this is as follows. If all the parts of a model are magnified in the same proportion, say m , and if thereby a line in it is increased in the ratio m:1 , then the areas and volumes in it will be increased respectively in the ratios m^2:1 and m^3:1. For example, if the side of a cube is doubled then a face of it will be increased in the ratio 4: 1 and its volume will be increased in the ratio 8:1.
Now if all the linear dimensions are increased m times, then some of the forces that act on a machine (such, for example, as the weight of part of it) will be increased m^3 times, while others which depend on area (such as the sustaining power of a beam) will be increased only m^2 times. Hence the forces that act on the machine and are brought into play by the various parts may be altered in different proportions, and thus the machine may be incapable of producing results similar to those which can be produced by the model.
The same argument has been adduced in the case of animal life to explain why very large specimens of any particular breed or species are usually weak. For example, if the linear dimensions of a bird were increased n times, the work necessary to give the power of flight would have to be increased no less than n^7 times (Helmholtz, Gesammelte Abhandlungen, Leipzig, 1881, vol. i, p. 165.). Again, if the linear dimensions of a man
of height 5 ft. 10 in. were increased by one-seventh his height would become 6 ft. 8 in., but his weight would be increased in the ratio 512: 343 ( about half as much again), while the cross sections of his legs, which would have to bear this weight, would be increased only in the ratio 64:49 ; thus in some respects he would be less efficient than before. Of course the increased dimensions, length of limb, or size of muscle might be of greater advantage than the relative loss of strength; hence the problem of what are the most efficient proportions is not simple, but the above argument will serve to illustrate the fact that the working of a machine may not be similar to that of a model of it.
Leaving now these elementary considerations I pass on to some other mechanical questions.
Sailing quicker than the Wind As a kinematical paradox I may allude to the possibility of sailing quicker than the wind blows, a fact which strikes many people as curious.
The explanation (Ozanam, 1803 edition, vol. iii, pp. 359, 367; 1840 edition, pp. 540, 543.) depends on the consideration of the velocity of the wind relative to the boat. Perhaps, however, a non-mathematician will find the solution simplified if I consider first the effect of the wind-pressure on the back of the sail which drives the boat forward, and second the resistance to motion caused by the sail being forced through the air.
When the wind is blowing against a plane sail the resultant pressure of the wind on the sail may be resolved into two components, one perpendicular to the sail (but which in general is not a function only of the component velocity in that direction, though it vanishes when that component vanishes) and the other parallel to its plane. The latter of these has no effect on the motion of the ship. The component perpendicular to the sail tends to move the ship in that direction. This pressure, normal to the sail, may be resolved again into two components, one in the direction of the keel of the boat,
the other in the direction of the beam of the boat. The former component drives the boat forward, the latter to leeward. It is the object of a boat-builder to construct the boat on lines so that the resistance of the water to motion forward shall be as small as possible, and the resistance to motion in a perpendicular direction ( to leeward) shall be as large as possible; and I will assume for the moment that the former of these resistances may be neglected, and that the latter is so large as to render motion in that direction impossible.
Now, as the boat moves forward, the pressure of the air on the front of the sail will tend to stop the motion. As long as its component normal to the sail is less than the pressure of the wind behind the sail and normal to it, the resultant of the two will be a force behind the sail and normal
to it which tends to drive the boat forwards. But as the velocity of the boat increases, a time will arrive when the pressure of the wind is only just able to balance the resisting force which is caused by the sail moving through the air. The velocity of the boat will not increase beyond this, and the motion will be then what mathematicians describe as "steady."
In the accompanying figure, let BAR represent the keel of a boat, B being the bow, and let SAL represent the sail. Suppose that the wind is blowing in the direction W!A with a velocity u ; and that this direction makes an angle θ with
the keel, angle W!AR = θ. Suppose that the sail is set so as to make an angle α with the keel, angle BAS = α , and therefore angle W!AL=θ + α. Suppose finally that v is the velocity of the boat in the direction AB.
I have already shown that the solution of the problem depends on the relative directions and velocities of the wind and the boat; hence to find the result reduce the boat to rest by impressing on it a velocity v in the direction BA. The resultant velocity of v parallel to BA and of u parallel to W!A will be parallel to SL , if v α = u (θ + α) ; and in this case the resultant pressure perpendicular to the sail vanishes.
Thus, for steady motion we have v α=u (θ + α). Hence, whenever (θ + α)> α , we have v>u. Suppose, to take one instance, the sail to be fixed, that is, suppose α to be a constant. Then v is a maximum if θ + α = 1/2π , that is, if θ is equal to the complement of α. In this case we have v = u α , and therefore v is greater than u. Hence, if the wind makes the same angle α abaft the beam that the sail makes with the keel, the velocity of the boat will be greater than the velocity of the wind.
Next, suppose that the boat is running close to the wind, so that the wind is before the beam (see figure Figure:112below),
then in the same way as before we have v α = u (θ + α) , or v α = u φ , where φ = angle W!AS = π - θ - α. Hence v = u φ α.
Let w be the component velocity of the boat in the teeth of the wind, that is, in the direction AW. Then we have w = v BAW = v (α + φ) = u φ α (α + φ). If α is constant, this is a maximum when φ = 1/4π - 1/2α ; and, if φ has this value, then w = 1/2u (α - 1). This formula shows that w is greater than u , if α< 1/3. Thus, if the sails can be set so that α is less than ^-11/3 , that is, rather less than 19^circ 29' , and if the wind has the direction above assigned, then the component velocity of the boat in the face of the wind is greater than the velocity of the wind.
The above theory is curious, but it must be remembered that in practice considerable allowance has to be made for the fact that no boat for use on water can be constructed in
which the resistance to motion in the direction of the keel can be wholly neglected, or which would not drift slightly to leeward if the wind was not dead astern. Still this makes less difference than might be thought by a landsman. In the case of boats sailing on smooth ice the assumptions made are substantially correct, and the practical results are said to agree closely with the theory.
[Boat moved by a rope inside the boat][Boat moved by a Rope.] Boat moved by a Rope There is a form of boat-racing, occasionally used at regattas, which affords a somewhat curious illustration of certain mechanical principles. The only thing supplied to the crew is a coil of rope, and they have, without leaving the boat, to propel it from one point to another as rapidly as possible. The motion is given by tying one end of the rope to the after thwart, and giving the other end a series of violent jerks in a direction parallel to the keel. I am told that in still water a pace of two or three miles an hour can be thus attained.
The chief cause for this result seems to be that the friction between the boat and the water retards all relative motion, but is not great enough to materially affect motion caused by a sufficiently big impulse. Hence the usual movements of the crew in the boat do not sensibly move the centre of gravity of themselves and the boat, but this does not apply to an impulsive movement, and if the crew in making a jerk move their centre of gravity towards the bow n times more rapidly than it returns after the jerk, then the boat is impelled forwards at least n times more than backwards: hence on the whole the motion is forwards.
[Results dependent on Hauksbee's Law][Motion of Fluids.] Motion of Fluids and Motion in Fluids The theories of motion of fluids and motion in fluids involve considerable difficulties. Here I will mention only one or two instances--mainly illustrations of Hauksbee's Law.
Hauksbee's Law When a fluid is in rapid motion the pressure is less than when it is at rest (See Besant, Hydromechanics, Cambridge, 1867, art. 149, where however it is assumed that the pressure is proportional to the density. Hauksbee was the earliest writer who called attention to the problem, but I do not know who first explained the phenomenon; some references to it are given by Willis, Cambridge Philosophical Transactions, 1830, vol. iii, pp. 129--140.). Thus, if a current of air is moving in a tube, the pressure on the sides of the tube is less than when the air is at rest--and the quicker the air moves the smaller is the pressure. This fact was noticed by Hauksbee nearly two centuries ago. In an elastic perfect fluid in which the pressure is proportional to the density, the law connecting the pressure, p , and the steady velocity, v , is p = Πα^(-v^2) where Π and α are constants: the establishment of corresponding formula for gases where the pressure is proportional to a power of the density presents no difficulty.
This principle is illustrated by a twopenny toy, on sale in most toy-shops, called the pneumatic mystery. It consists of a tube, with a cup-shaped end in which rests a wooden ball. If the tube is held in a vertical position, with the mouthpiece at
the upper end and the cup at the lower end, then, if anyone blows hard through the tube and places the ball against the cup, the ball will remain suspended there. The explanation is that the pressure of the air below the ball is so much greater than the pressure of the air in the cup that the ball is held up.
The same effect may be produced by fastening to one end of a tube a piece of cardboard having a small hole in it. If a piece of paper is placed over the hole and the experimenter blows through the tube, the paper will not be detached from the card but will bend so as to allow the egress of the air.
An exactly similar experiment, described in many text-books on hydromechanics, is made as follows. To one end of a straight tube a plane disc is fitted which is capable of sliding on wires projecting from the end of the tube. If the disc is placed at a small distance from the end, and anyone blows steadily into the tube, the disc will be drawn towards the tube instead of being blown off the wires, and will oscillate about a position near the end of the tube.
In the same way we may make a tube by placing two books on a table with their backs parallel and an inch or so apart and laying a sheet of newspaper over them. If anyone blows steadily through the tube so formed, the paper will be sucked in instead of being blown out.
The following experiment is explicable by the same argument. On the top of a vertical axis balance a thin horizontal rod. At each end of this rod fasten a small vertical square or sail of thin cardboard--the two sails being in the same plane. If anyone blows close to one of these squares and in a direction parallel to its plane, the square will move towards the side on which one is blowing, and the rod with the two sails will rotate about the axis.
The experiments above described can be performed so as to illustrate Hauksbee's Law; but unless care is taken other causes will be also introduced which affect the phenomena: it is however unnecessary for my purpose to go into these details.
[Cut on a tennis-ball. Spin on a cricket-ball] [Spin on Tennis and Cricket Balls.]Cut on a Tennis-Ball Racquet and tennis players know that if a strong cut is given to a ball it can be made to
rebound off a vertical wall and then (without striking the floor or any other wall) return and hit the wall again.
This affords another illustration of Hauksbee's Law. The explanation (See Magnus on 'Die Abweichung der Geschosse' in the Abhandlungen der Akademie der Wissenschaften, Berlin, 1852, pp. 1--23; Lord Rayleigh, 'On the irregular flight of a tennis ball,' Messenger of Mathematics, Cambridge, 1878, vol. vii, pp. 14--16.) is that the cut causes the ball to rotate rapidly about an axis through its centre of figure, and the friction of the surface of the ball on the air produces a sort of whirlpool. This rotation is in addition to its motion of translation. Suppose the ball to be spherical and rotating about an axis through its centre perpendicular to the plane of the paper in the direction of the arrow-head, and at the same time moving through still air from left to right parallel to PQ. Any motion of the ball perpendicular to PQ will be produced by the pressure of the air on the surface of the ball, and this pressure will, by Hauksbee's Law, be greatest where the velocity of the air relative to the ball is least, and vice versa. To find the velocity of the air relative to the ball we may reduce the centre of the ball to rest, and suppose a stream of air to impinge on the surface of the ball moving with a velocity equal and opposite to that of the centre of the ball. The air is not frictionless, and therefore the air in contact with the surface of the ball will be set in motion, by the rotation of the ball and will form a sort of whirlpool rotating in the direction of the arrow-head in the figure. To find the actual velocity of this air relative to the ball we must consider how the motion due to the whirlpool is affected by the motion of the stream of air parallel to QP. The air at A in the whirlpool is moving against the stream of air there, and therefore its velocity is retarded: the air at B in the whirlpool is moving in the same direction as the stream of air there, and therefore its velocity is increased. Hence the relative velocity of the air at A is less than that at B , and
since the pressure of the air is greatest where the velocity is least, the pressure of the air on the surface of the ball at A is greater than on that at B , Hence the ball is forced by this
pressure in the direction from the line PQ , which we may suppose to represent the section of the vertical wall in a racquet-court. In other words, the ball tends to move at right angles to the line in which its centre is moving and in the direction in which the surface of the front of the ball is being carried by the rotation.
In the case of a lawn tennis-ball, the shape of the ball is altered by a strong cut, and this introduces additional complications
.
Spin on a Cricket-Ball The curl of a cricket-ball in its flight through the air, caused by a spin given by the bowler in delivering the ball, is explained by the same reasoning.
Thus suppose the ball is delivered in a direction lying in a vertical plane containing the two middle stumps of the wickets. A spin round a horizontal axis parallel to the crease in a direction which the bowler's umpire would describe as positive, namely, counter clock-wise, will, in consequence of the friction of the air, cause it to drop, and therefore decrease the length of the pitch. A spin in the opposite direction will cause it to rise, and therefore lengthen the pitch. A spin round a vertical axis in the positive direction, as viewed from above, will make it curl sideways in the air to the left, that is, from leg to off. A spin in the opposite direction will make it curl to the right. A spin given to the ball round the direction of motion of the
centre of the ball will not sensibly affect the motion through the air, though it would cause the ball, on hitting the ground, to break. Of course these various kinds of spin can be combined.
The questions involving the application of Hauksbee's Law are easy as compared with many of the problems in fluid motion. The analysis required to attack most of these problems is beyond the scope of this book, but one of them may be worth mentioning even though no explanation is given.
The Theory of the Flight of Birds Flight of Birds Flight of Birds. A mechanical problem of great interest is the explanation of the means by which birds are enabled to fly for considerable distances with no (perceptible) motion of the wings. Albatrosses, to take an instance of special difficulty, have been known to follow for some days ships running at the rate of nine or ten knots, and sometimes for considerable periods there is no motion of the wings or body which can be detected, while even if the bird moved its wings it is not easy to understand how it has the muscular energy to propel itself so rapidly and for such a length of time. Of this phenomenon various explanations (See G.H. Bryan in the Transactions of the British Association for 1896, vol. lxvi, pp. 726-728.) have been suggested. Notable among these are Mr Maxim's of upward air-currents, Lord Rayleigh's of variations of the wind velocity at different heights above the ground, Dr S.P. Langley's of the incessant occurrence of gusts of wind separated by lulls, and Dr Bryan's of vortices in the atmosphere.
It now seems reasonably certain that the second and third of these sources of energy account for at least a portion of the observed phenomena. The effect of the third cause may be partially explained by noting that the centre of gravity of the bird with extended wings is slightly below the aeroplane or wing surface, so that the animal forms a sort of parachute. The effect of a sudden gust of wind upon such a body is that the aeroplane is set in motion more rapidly than the suspended mass, causing
the structure to heel over so as to receive the wind on the under surface of the aeroplane, and this lifts the suspended mass giving it an upward velocity. When the wind falls the greater inertia of the mass carries it on upwards causing the aeroplane to again present its under side to the air; and if while the parachute is in this position the wind is still blowing from the side, the suspended mass is again lifted. Thus the more the bird is blown about, the more it rises in the air; actually birds in flight are carried up by a sudden side gust of wind as we should expect from this theory.
The fact that the bird is in motion tends also to keep it up, for it has been recently shown that a horizontal plane under the action of gravity falls to the ground more slowly if it is travelling through the air with horizontal velocity than it would do if allowed to fall vertically, hence the bird's forward motion causes it to fall through a smaller height between successive gusts of wind than it would do if it were at rest, Moreover it has been proved experimentally that the horsepower required to support a body in horizontal flight by means of an aeroplane is less for high than for low speeds: hence when a side-wind (that is, a wind at right angles to the bird's course) strikes the bird, the lift is increased in consequence of the bird's forward velocity.
When I was writing the first edition of these "Recreations," I put together a chapter, following this one, on "Some Physical Questions," dealing with problems such as, in the Theory of Sound, the explanation of the fact that in some of Captain Parry's experiments the report of a cannon, when fired, travelled so much more rapidly than the sound of the human voice that observers heard the report of the cannon when fired before that of the order to fire it (The fact is well authenticated. Mr Earnshaw (Philosophical Transactions, London, 1860, pp. 133--148) explained it by the acceleration of a wave caused by the formation of a kind of bore, a view accepted by Clerk Maxwell and most physicists, but Sir George Airy thought that the explanation was to be found in physiology; see Airy's Sound, second edition, London, 1871, pp. 141, 142.): in the Kinetic Theory of Gases, the complications in
our universe that might be produced by "Maxwell's demon" (See Theory of Heat, by J. Clerk Maxwell, second edition, London, 1872, p. 308.): in the Theory of Optics, the explanation of the Japanese "magic mirrors," (See a memoir by W.E. Ayrton and J. Perry, Proceedings of the Royal Society of London, part i, 1879, vol. xxviii, pp. 127--148.) which reflect the pattern on the back of the mirror (on which the light does not fall): to which I might add the theory of the "spectrum top," by means of which a white surface, on which some black lines are drawn, can be moved so as to give the impression (See letters from Mr C.E. Benham and others in Nature, 1894--5; and a memoir by Prof. Liveing, Cambridge Philosophical Society, November 26, 1894.) that the lines are coloured (red, green, blue, slate, or drab), and the curious fact that the colours change with the direction of rotation: it has also been recently shown that if two trains of waves, whose lengths are in the ratio m-1: m+1 , be superposed, then every m th wave in the system will be big--thus the current opinion that every ninth wave in the open sea is bigger than the other waves may receive scientific confirmation. There is no lack of interesting and curious phenomena in physics, and in some branches, notably in electricity and magnetism, the difficulty is rather one of selection, but I felt that the connection with mathematics was in general either too remote or too technical to justify the insertion of such a collection in a work on elementary mathematical recreations, and therefore I struck out the chapter. I mention the fact now partly to express the hope that some physicist will one day give us a collection of the kind, partly to suggest these questions to those who are interested in such matters.
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End of part 5 of 16
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