Part 4

A wooden rod, A B 48" × 0"·5 × 0"·5, has strings attached to it at points A and D, one foot distant. The string at D passes over a fixed pulley E, and at the end P a hook is attached for the purpose of receiving weights, while a similar hook descends from A; the weight of the rod itself, which only amounts to three ounces, may be neglected, as it is very small compared with the weights which will be used.

73. Supposing 2 lbs. to be placed at P, and 1 lb. at Q, we have two parallel forces acting in opposite directions; and since their difference is 1 lb., it follows from our rule that the point F, where D F is equal to A D, is the point where the resultant is applied. You see this is easily verified, for by placing my finger over the rod at F it remains horizontal and in equilibrium; whereas, when I move my finger to one side or the other, equilibrium is impossible. If I move it nearer to B, the end A ascends. If I move it towards A, the end B ascends.

74. To study the case when the two forces are equal, a load of 2 lbs. may be placed on each of the hooks P and Q. It will then be found that the finger cannot be placed in any position along the rod so as to keep it in equilibrium; that is to say, no single force can counteract the two forces which form the couple. Let O be the point midway between A and D. The forces evidently tend to raise OB and turn the part O A downwards; but if I try to restrain O B by holding my finger above, as at the point X, instantly the rod begins to turn round X and the part from A to X descends. I find similarly that any attempt to prevent the motion by holding my finger underneath is equally unsuccessful. But if at the same time I press the rod downwards at one point, and upwards at another with suitable force, I can succeed in producing equilibrium; in this case the two pressures form a couple; and it is this couple which neutralizes the couple produced by the weights. We learn, then, the important result that _a couple can be balanced by a couple_, and by a couple only.

75. The moment of a couple is the product of one of the two equal forces into their perpendicular distance. Two couples tending to turn the body to which they are applied in the _same_ direction will be equivalent if their moments are equal. Two couples which tend to turn the body in _opposite_ directions will be in equilibrium if their moments are equal. We can also compound two couples in the same or in opposite directions into a single couple of which the moment is respectively either the sum or the difference of the original moments.

THE WEIGHING SCALES.

76. Another apparatus by which the nature of parallel forces may be investigated is shown in Fig. 24; it consists of a slight frame of wood A B C, 4' long. At E, a pair of steel knife-edges is clamped to the frame. The knife-edges rest on two pieces of steel, one of which is shown at O F. When the knife-edges are suitably placed the frame is balanced, so that a small piece of paper laid at A will cause that side to descend.

77. We suspend two small hooks from the points A and B: these are made of fine wire, so that their weight may be left out of consideration. With this apparatus we can in the first place verify the principle of equality of moments: for example, if I place the hook A at a distance of 9" from the centre O and load it with 1 lb., I find that when B is laden with 0·5 lb. it must be at a distance of 18" from O in order to counterbalance A; the moment in the one case is 9 × 1, in the other 18 × 0·5, and these are obviously equal.

78. Let a weight of 1 lb. be placed on each of the hooks, the frame will only be in equilibrium when the hooks are at precisely the same distance from the centre. A familiar application of this principle is found in the ordinary weighing scales; the frame, which in this case is called a _beam_, is sustained by two knife-edges, smaller, however, than those represented in the figure. The pans P, P are suspended from the extremities of the beam, and should be at equal distances from its centre. These scale-pans must be of equal weight, and then, when equal weights are placed in them, the beam will remain horizontal. If the weight in one slightly exceed that in the other, the pan containing the heavier weight will of course descend.

79. That a pair of scales should weigh accurately, it is necessary that the weights be correct; but even with correct weights, a balance of defective construction will give an inaccurate result. The error frequently arises from some inequality in the lengths of the arms of the beam. When this is the case, the two weights which really balance are not equal. Supposing, for instance, that with an imperfect balance I endeavour to weigh a pound of shot. If I put the weight on the short side, then the quantity of shot balanced is less than 1 lb.; while if the 1 lb. weight be placed at the long side, it will require more than 1 lb. of shot to produce equilibrium. The mode of testing a pair of scales is then evident. Let weights be placed in the pans which balance each other; if the weights be interchanged and the balance still remains horizontal, it is correct.

80. Suppose, for example, that the two arms be 10 inches and 11 inches long, then, if 1 lb. weight be placed in the pan of the 10-inch end, its moment is 10; and if ¹⁰/₁₁ of 1 lb. be placed in the pan belonging to the 11-inch end, its moment is also 10: hence 1 lb. at the short end balances ¹⁰/₁₁ of 1 lb. at the long end; and therefore, if the shopkeeper placed his weight in the short arm, his customers would lose ¹/₁₁ part of each pound for which they paid; on the other hand, if the shopkeeper placed his 1 lb. weight on the long arm, then not less than ¹¹/₁₀ lb. would be required in the pan belonging to the short arm. Hence in this case the customer would get ¹/₁₀ lb. too much. It follows, that if a shopman placed his weights and his goods alternately in the one scale and in the other he would be a loser on the whole; for, though every second customer gets ¹/₁₁ lb. less than he ought, yet the others get ¹/₁₀ lb. more than they have paid for.

LECTURE IV. _THE FORCE OF GRAVITY._

Introduction.—Specific Gravity.—The Plummet and Spirit-Level.—The Centre of Gravity.—Stable and Unstable Equilibrium.—Property of the Centre of Gravity in a Revolving Wheel.

INTRODUCTION.

81. In the last three lectures we considered forces in the abstract; we saw how they are to be represented by straight lines, how compounded together and how decomposed into others; we have explained what is meant by forces being in equilibrium, and we have shown instances where the forces lie in the same plane or in different planes, and where they intersect or are parallel to each other. These subjects are the elements of mechanics; they form the framework which in this and subsequent lectures we shall try to present in a more attractive garb. We shall commence by studying the most remarkable force in nature, a force constantly in action, and one to which all bodies are subject, a force which distance cannot annihilate, and one the properties of which have led to the most sublime discoveries of human intellect. This is the force of gravity.

82. If I drop a stone from my hand, it falls to the ground. That which produces motion is a force: hence the stone must have been acted upon by a force which drew it to the ground. On every part of the earth’s surface experience shows that a body tends to fall. This fact proves that there is an attractive force in the earth tending to draw all bodies towards it.

83. Let A B C D (Fig. 25) be points from which stones are let fall, and let the circle represent the section of the earth; let P Q R S be the points at the surface of the earth upon which the stones will drop when allowed to do so. The four stones will move in the directions of the arrows: from A to P the stone moves in an opposite direction to the motion from C to R; from B to Q it moves from right to left, while from D to S it moves from left to right. The movements are in different directions; but if I produce these directions, as indicated by the dotted lines, they each pass through the centre O.

84. Hence each stone in falling moves towards the centre of the earth, and this is consequently the direction of the force. We therefore assert that the earth has an attraction for the stone, in consequence of which it tries to get as near the earth’s centre as possible, and this attraction is called the force of gravitation.

85. We are so excessively familiar with the phenomenon of seeing bodies fall that it does not excite our astonishment or arouse our curiosity. A clap of thunder, which every one notices, because much less frequent, is not really more remarkable. We often look with attention at the attraction of a piece of iron by a magnet, and justly so, for the phenomenon is very interesting, and yet the falling of a stone is produced by a far grander and more important force than the force of magnetism.

86. It is gravity which causes the weight of bodies. I hold a piece of lead in my hand: gravity tends to pull it downwards, thus producing a pressure on my hand which I call _weight_. Gravity acts with slightly varying intensity at various parts of the earth’s surface. This is due to two distinct causes, one of which may be mentioned here, while the other will be subsequently referred to. The earth is not perfectly spherical; it is flattened a little at the poles; consequently a body at the pole is nearer the general mass of the earth than a body at the equator; therefore the body at the pole is more attracted, and seems heavier. A mass which weighs 200 lbs. at the equator would weigh one pound more at the pole: about one-third of this increase is due to the cause here pointed out. (See Lecture XVII.)

87. Gravity is a force which attracts every particle of matter; it acts not merely on those parts of a body which lie on the surface, but it equally affects those in the interior. This is proved by observing that a body has the same weight, however its shape be altered: for example, suppose I take a ball of putty which weighs 1 lb., I shall find that its weight remains unchanged when the ball is flattened into a thin plate, though in the latter case the surface, and therefore the number of superficial particles, is larger than it was in the former.

SPECIFIC GRAVITY.

88. Gravity produces different effects upon different substances. This is commonly expressed by saying that some substances are heavier than others; for example, I have here a piece of wood and a piece of lead of equal bulk. The lead is drawn to the earth with a greater force than the wood. Substances are usually termed heavy when they sink in water, and light when they float upon it. But a body sinks in water if it weighs more than an equal bulk of water, and floats if it weigh less. Hence it is natural to take water as a standard with which the weights of other substances may be compared.

89. I take a certain volume, say a cubic inch of cast iron such as this I hold in my hand, and which has been accurately shaped for the purpose. This cube is heavier than one cubic inch of water, but I shall find that a certain quantity of water is equal to it in weight; that is to say, a certain number of cubic inches of water, and it may be fractional parts of a cubic inch, are precisely of the same weight. This number is called the _specific gravity of cast iron_.

90. It would be impossible to counterpoise water with the iron without holding the water in a vessel, and the weight of the vessel must then be allowed for. I adopt the following plan. I have here a number of inch cubes of wood (Fig. 26), which would each be lighter than a cubic inch of water, but I have weighted the wooden cubes by placing grains of shot into holes bored into the wood. The weight of each cube has thus been accurately adjusted to be equal to that of a cubic inch of water. This may be tested by actual weighing. I weigh one of the cubes and find it to be 252 grains, which is well known to be the weight of a cubic inch of water.

91. But the cubes may be shown to be identical in weight with the same bulk of water by a simpler method. One of them placed in water should have no tendency to sink, since it is not heavier than water, nor on the other hand, since it is not lighter, should it have any tendency to float. It should then remain in the water in whatever position it may be placed. It is difficult to prepare one of these cubes so accurately that this result should be attained, and it is impossible to ensure its continuance for any time owing to changes of temperature and the absorption of water by the wood. We can, however, by a slight modification, prove that one of these cubes is at all events nearly equal in weight to the same bulk of water. In Fig. 26 is shown a tall glass jar filled with a fluid in appearance like plain water, but it is really composed in the following manner. I first poured into the jar a very weak solution of salt and water, which partially filled it; I then poured gently upon this a little pure water, and finally filled up the jar with water containing a little spirits of wine: the salt and water is a little heavier than pure water, while the spirit and water is a little lighter. I take one of the cubes and drop it gently into the glass; it falls through the spirit and water, and after making a few oscillations settles itself at rest in the stratum shown in the figure. This shows that our prepared cube is a little heavier than spirit and water, and a little lighter than salt and water, and hence we infer that it must at all events be very near the weight of pure water which lies between the two. We have also a number of half cubes, quarter cubes, and half-quarter cubes, which have been similarly prepared to be of equal weight with an equal bulk of water.

92. We shall now be able to measure the specific gravity of a substance. In one pan of the scales I place the inch cube of cast iron, and I find that 7¼ of the wooden cubes, which we may call cubes of water, will balance it. We therefore say that the specific gravity of iron is 7¼. The exact number found by more accurate methods is 7·2. It is often convenient to remember that 23 cubic inches of cast iron weigh 6 lbs., and that therefore one cubic inch weighs very nearly ¼ lb.

93. I have also cubes of brass, lead, and ivory; by counterpoising them with the cubes of water, we can easily find their specific gravities; they are shown together with that of cast iron in the following table:—

Substance. Specific Gravity. Cast Iron 7·2 Brass 8·1 Lead 11·3 Ivory 1·8

94. The mode here adopted of finding specific gravities is entirely different from the far more accurate methods which are commonly used, but the explanation of the latter involve more difficult principles than those we have been considering. Our method rather offers an explanation of the nature of specific gravity than a good means of determining it, though, as we have seen, it gives a result sufficiently near the truth for many purposes.

THE PLUMMET AND SPIRIT-LEVEL.

95. The tendency of the earth to draw all bodies towards it is well illustrated by the useful “line and plummet.” This consists merely of a string to one end of which a leaden weight is attached. The string when at rest hangs vertically; if the weight be drawn to one side, it will, when released, swing backwards and forwards, until it finally settles again in the vertical; the reason is that the weight always tries to get as near the earth as it can, and this is accomplished when the string hangs vertically downwards.

96. The surface of water in equilibrium is a horizontal plane; that is also a consequence of gravity. All the particles of water try to get as near the earth as possible, and therefore if any portion of the water were higher than the rest, it would immediately spread, as by doing so it could get lower.

97. Hence the surface of a fluid at rest enables us to find a perfectly horizontal plane, while the plummet gives us a perfectly vertical line: both these consequences of gravity are of the utmost practical importance.

98. The spirit-level is another common and very useful instrument which depends on gravity. It consists of a glass tube slightly curved, with its convex surface upwards, and attached to a stand with a flat base. This tube is nearly filled with spirit, but a bubble of air is allowed to remain. The tube is permanently adjusted so that, when the plate is laid on a perfectly horizontal surface, the bubble will stand in the middle: accordingly the position of the bubble gives a means of ascertaining whether a surface is level.

THE CENTRE OF GRAVITY.

99. We proceed to an experiment which will give an insight into a curious property of gravity. I have here a plate of sheet iron; it has the irregular shape shown in Fig. 27. Five small holes A B C D E are punched at different positions on the margin. Attached to the framework is a small pin from which I can suspend the iron plate by one of its holes A: the plate is not supported in any other way; it hangs freely from the pin, around which it can be easily turned. I find that there is one position, and one only, in which the plate will rest; if I withdraw it from that position it returns there after a few oscillations. In order to mark this position, I suspend a line and plummet from the pin, having rubbed the line with chalk. I allow the line to come to rest in front of the plate. I then flip the string against the plate, and thus produce a chalked mark: this of course traces out a vertical line A P on the plate.

I now remove the plummet and suspend the plate from another of its holes B, and repeat the process, thus drawing a second chalked line B P across the plate, and so on with the other holes: I thus obtain five lines across the plate, represented by dotted lines in the figure. It is a very remarkable circumstance that these five lines all intersect in the same point P; and if additional holes were bored in the plate, whether in the margin or not, and the chalk line drawn from each of them in the manner described, they would one and all pass through the same point. This remarkable point is called _the centre of gravity_ of the plate, and the result at which we have arrived may be expressed by saying that the vertical line from the point of suspension always passes through the centre of gravity.

100. At the centre of gravity P a hole has been bored, and when I place the supporting pin through this hole you see that the plate will rest indifferently in all positions: this is a curious property of the centre of gravity. The centre of gravity may in this respect be contrasted with another hole Q, which is only an inch distant: when I support the plate by this hole, it has only one position of rest, viz. when the centre of gravity P is vertically beneath Q. Thus the centre of gravity differs remarkably from any other point in the plate.

101. We may conceive the force of gravity on the plate to act as a force applied at P. It will then be easily seen why this point remains vertically underneath the point of suspension when the body is at rest. If I attached a string to the plate and pulled it, the plate would evidently place itself so that the direction of the string would pass through the point of suspension; in like manner gravity so places the plate that the direction of its force passes through the point of suspension.

102. Whatever be the form of the plate it always contains one point possessing these remarkable properties, and we may state in general that in every body, no matter what be its shape, there is a point called the centre of gravity, such that if the body be suspended from this point it will remain in equilibrium indifferently in any position, and that if the body be suspended from any other point, then it will be in equilibrium when the centre of gravity is directly underneath the point of suspension. In general, it will be impossible to support a body exactly at its centre of gravity, as this point is within the mass of the body, and it may also sometimes happen that the centre of gravity does not lie in the substance of the body at all, as for example in a ring, in which case the centre of gravity is at the centre of the ring. We need not, however, dwell on these exceptional cases, as sufficient illustrations of the truth of the laws mentioned will present themselves subsequently.

STABLE AND UNSTABLE EQUILIBRIUM.

103. An iron rod A B, capable of revolving round an axis passing through its centre P, is shown in Fig. 28.

The centre of gravity lies at the centre B, and consequently, as is easily seen, the rod will remain at rest in whatever position it be placed. But let a weight R be attached to the rod by means of a binding screw. The centre of gravity of the whole is no longer at the centre of the rod; it has moved to a point S nearer the weight; we may easily ascertain its position by removing the rod from its axle and then ascertaining the point about which it will balance. This may be done by placing the bar on a knife-edge, and moving it to and fro until the right position be secured; mark this position on the rod, and return it to its axle, the weight being still attached. We do not now find that the rod will balance in every position. You see it will balance if the point S be directly underneath the axis, but not if it lie to one side or the other. But if S be directly over the axis, as in the figure, the rod is in a curious condition. It will, when carefully placed, remain at rest; but if it receive the slightest displacement, it will tumble over. The rod is in equilibrium in this position, but it is what is called _unstable_ equilibrium. If the centre of gravity be vertically below the point of suspension, the rod will return again if moved away: this position is therefore called one of _stable_ equilibrium. It is very important to notice the distinction between these two kinds of equilibrium.

104. Another way of stating the case is as follows. A body is in stable equilibrium when its centre of gravity is at the lowest point: unstable when it is at the highest. This may be very simply illustrated by an ellipse, which I hold in my hand. The centre of gravity of this figure is at its centre. The ellipse, when resting on its side, is in a position of stable equilibrium and its centre of gravity is then clearly at its lowest point. But I can also balance the ellipse on its narrow end, though if I do so the smallest touch suffices to overturn it. The ellipse is then in unstable equilibrium; in this case, obviously, the centre of gravity is at the highest point.

105. I have here a sphere, the centre of gravity of which is at its centre; in whatever way the sphere is placed on a plane, its centre is at the same height, and therefore cannot be said to have any highest or lowest point; in such a case as this the equilibrium is _neutral_. If the body be displaced, it will not return to its old position, as it would have done had that been a position of stable equilibrium, nor will it deviate further therefrom as if the equilibrium had been unstable: it will simply remain in the new position to which it is brought.

106. I try to balance an iron ring upon the end of a stick H, Fig. 29, but I cannot easily succeed in doing so. This is because its centre of gravity S is above the point of support; but if I place the stick at F, the ring is in stable equilibrium, for now the centre of gravity is below the point of support.