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[PDF page 89] CH. III] MAGIC BOTTLES. 73 Magic Bottles*. 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 byC, and the centre of gravity of the bottle by G. Ifsuchabottleisplacedwiththehemisphererestingonahorizontal 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 atG. 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 aboveC, 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. Inasimilarwaywemayexplainhowtobalanceapencilinavertical position, with its point resting on the top of one’s finger, an experiment which is described in nearly every book of puzzles†. This is effected by taking a penknife, of which one blade is opened through an angle of (say) 120◦, 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. * Ozanam, 1803 edition, vol. ii, p. 15; 1840 edition, p. 201. † Ex. gr.Oughtred,Mathematical Recreations, p.24; Ozanam, 1803edition, vol.ii, p. 14; 1840 edition, p. 200.

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[PDF page 90] 74 MECHANICAL RECREATIONS. [CH. III 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*—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 down- wards 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 cen- tre 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†. 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 tothetopofa T-shapedwireweightedatitsends. Ifthefigureisplaced on a narrow sloping plank or strip of wood passing between the legs of theT, then owing to theT-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.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 * Oughtred, p. 30; Ozanam, 1803 edition, vol.ii, p. 12; 1840 edition, p. 199. † La Nature, Paris, March, 1891.

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[PDF page 91] CH. III] CONES RUNNING UP HILL. 75 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*. 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 W, 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 theW, 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 occu- pies 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 de- scends. Perpetual Motion. 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. Nowtheobviousmeaningofthethirdlawofmotionisthataforceis 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 interpreta- tion†, 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 * Ozanam, 1803 edition, vol. ii, p. 49; 1840 edition, p. 216. † Newton’sPrincipia, last paragraph of the Scholium to the Laws of Motion.

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[PDF page 92] 76 MECHANICAL RECREATIONS. [CH. III that done against molecular forces, then it follows from this interpre- tation 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. C 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 throughC than those on the left. Hence the sum of the moments of the weights of the bullets on the

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[PDF page 93] CH. III] PERPETUAL MOTION. 77 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 bobuptothesameheightineachswingandthusgiveperpetualmotion. 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, perpet- ual 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*. The energy which was stored up in it to maintain the motion of the pendulumwasprovidedbytheexpansionofasilverrod. Thisexpansion 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 * Ozanam, 1803 edition, vol. ii, p. 105; 1840 edition, p. 238.

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[PDF page 94] 78 MECHANICAL RECREATIONS. [CH. III up for twenty-four hours. Models. I may add here the observation, which is well known to mathematicians, but is a perpetual source of disappointment to igno- rant 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, saym, and if thereby a line in it is increased in the ratiom : 1, then the areas and volumes in it will be increased respectively in the ratiosm2 : 1 and m3 : 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 ratio8 : 1. Now if all the linear dimensions are increasedm times, then some of the forces that act on a machine (such, for example, as the weight of part of it) will be increasedm3 times, while others which depend on area (such as the sustaining power of a beam) will be increased only m2 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 thann7 times*. Again, if the linear di- mensions of a man of height5 ft. 10 in. were increased by one-seventh his height would become6 ft. 8 in., but his weight would be increased in the ratio512 : 343 (i.e. 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 ratio64 : 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 pro- portions 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. * Helmholtz, Gesammelte Abhandlungen, Leipzig, 1881, vol.i, p. 165.

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[PDF page 95] CH. III] SAILING QUICKER THAN THE WIND. 79 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 ofsailing quicker than the wind blows, a fact which strikes many people as curious. The explanation* 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 pres- sure 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 lat- ter 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 compo- nents, 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 (i.e. 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 * Ozanam, 1803 edition, vol. iii, pp. 359, 367; 1840 edition, pp. 540, 543.

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[PDF page 96] 80 MECHANICAL RECREATIONS. [CH. III BR S L A W increase beyond this, and the motion will be then what mathematicians describe as “steady.” In the accompanying figure, letBAR represent the keel of a boat, B being the bow, and let SAL represent the sail. Suppose that the wind is blowing in the directionWA with a velocity u; and that this direction makes an angleθ with the keel,i.e. angle WAR = θ. Suppose that the sail is set so as to make an angleα with the keel,i.e. angle BAS = α, and therefore angleWAL = θ + α. Suppose finally that v is the velocity of the boat in the directionAB. 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 directionBA. The resultant velocity ofv parallel to BA and of u parallel to WA will be parallel toSL, if v sin α = u sin(θ + α); and in this case the resultant pressure perpendicular to the sail vanishes. Thus, for steady motion we havev sin α = u sin(θ + α). Hence, whenever sin(θ + α) > sin α, we have v > u . Suppose, to take one instance, the sail to be fixed, that is, supposeα to be a constant. Then v is a maximum ifθ + α = 1 2 π, that is, ifθ is equal to the complement of α. In this case we havev = u cosec α, and thereforev 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 below), then in the same way as before we have v sin α = u sin(θ + α), or v sin α = u sin φ, where

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[PDF page 97] CH. III] BOAT MOVED BY A ROPE.. 81 φ = angle WAS = π − θ − α. Hence v = u sin φ cosec α. Let w be the component velocity of the boat in the teeth of the wind, that is, in the directionAW. Then we havew = v cos BAW = v cos(α + φ) = u sin φ cosec α cos(α + φ). If α is constant, this is a maximum when φ = 1 4 π − 1 2 α; and, if φ has this value, then w = 1 2 u(cosec α − 1). This formula shows that w is greater than u, if sin α < 1 3. Thus, if the sails can be set so that α is less than sin−1 1 3, that is, rather less than19◦29′, and if the wind has the direc- tion 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 BR S L A W 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. There is a form of boat-racing, occa- sionally used at regattas, which affords a somewhat curious illustration of certain mechanical principles. The only thing supplied to the crew

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[PDF page 98] 82 MECHANICAL RECREATIONS. [CH. III 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 bown times more rapidly than it returns after the jerk, then the boat is impelled forwards at leastn times more than backwards: hence on the whole the motion is forwards. Motion of Fluids and Motion in Fluids. The theories of motion of fluidsand motion in fluidsinvolve 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*. 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 noticedbyHauksbeenearlytwocenturiesago. Inanelasticperfectfluid in which the pressure is proportional to the density, the law connecting the pressure, p, and the steady velocity,v, is p = Πα−v2 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 thepneumatic 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 * 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.

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[PDF page 99] CH. III] SPIN ON TENNIS AND CRICKET BALLS.. 83 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 illus- trate 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. 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 expla- nation* is that the cut causes the ball to rotate rapidly about an axis * See Magnus on ‘Die Abweichung der Geschosse ’ in theAbhandlungen der
