CHAPTER VI.

CENTRE OF GRAVITY.

95. =Centre of Gravity Illustrated.=--If you balance a ruler on your finger, as in Fig. 30, it is balanced because there is just as much weight on one side as on the other. Now just over your finger, in the middle of the ruler, there is a point that we call the centre of gravity; or, in other words, the centre of the weight of the ruler. This point is indicated in the figure. There is as much of the weight of the ruler on the one side of this point as on the other, and also as much above it as below it. If your finger should be a little to the one side or the other of this point, the ruler would not be balanced, and would fall. When balanced it does not fall, simply because this central point is supported by being directly over the end of the finger. The weight of the ruler, then, may be considered practically as all being in that point, for it is there that is exerted all the downward pressure of the ruler as it is balanced. So, also, when the ruler is balanced on the finger, as represented in Fig. 31 (p. 68), this same centre of gravity is directly over the point of the finger, and is therefore supported. If it be to the one side or the other, as in Fig. 32, it is not supported, and the ruler therefore falls. You see, then, that if a body be balanced, the centre of gravity is directly _over_ the point of support. If, on the other hand, a body is suspended, the centre of gravity is directly _under_ the point of support.

96. =Definition.=--If a plumb-line from the centre of gravity could be extended into the earth it would go directly to its centre. The body may be considered as making all its pressure from its centre of gravity in that direction, in obedience to the attraction of gravitation. The best definition, then, that we can give of the centre of gravity is, that it is _that point in a body from which its pressure as a whole toward the centre of the earth proceeds_. It is that point, therefore, the support of which insures the support of the whole body. And in speaking of the weight of a body, or its downward pressure, we may consider all the matter composing it as collected or concentrated in that point. The body, therefore, can be balanced in any position in which this point is supported, as shown in Figs. 30 and 31. And when a body is suspended, it is at rest only when the centre of gravity is directly under the point of support. Thus, if you have a circular plate suspended at E, Fig. 33, it will not be at rest when it is moved to the one side or the other, as represented by the dotted lines, but only when the centre of gravity, _c_, is directly under the point E.

97. =How to Find the Centre of Gravity of a Body.=--If you take a piece of board, and suspend it at A, Fig. 34, and suspend a plumb-line from the same point, the centre must be somewhere in that line. But exactly at what point it is you do not know. How will you ascertain this? Mark the line A B on the board, and suspend the board by another point, as in Fig. 35. As the centre of gravity must be somewhere in the plumb-line as it now hangs, of course it is at O, where the two lines cross.

98. =Scales and Steelyards.=--When two bodies are connected by a rod or bar, the centre of gravity of the whole is somewhere in the connection. If the two bodies be equal in weight, as in Fig. 36, the centre of gravity is exactly in the middle of the rod, as marked. But if the bodies are unequal, as in Fig. 37, the centre of gravity is nearer to the larger body than to the smaller. In balancing a body in one scale with weights in another, you have a case parallel to that of Fig. 36. The centre of gravity of the body weighed, the weights, and the scales, as a whole, is midway between the scales, at the point of support. In the steelyard you have the heavy body to be weighed nearer the centre of gravity than the small weight is on the long arm, and so the case is parallel with that of Fig. 37.

99. =The Centre of Gravity of a Body not Always in the Body Itself.=--The centre of gravity of a hollow ball of uniform thickness is not in the substance of the ball, but it is in the centre of the space in the ball, for the line of the ball's downward pressure would be from that point. If the ball had a frame-work in it, as represented in Fig. 38, the centre of gravity would obviously be at A, the centre of this frame-work. But if there were no frame-work, and perpendicular lines were supposed to be drawn from different points of suspension, C, B, D, and E, these would intersect at the point A, showing that this is the centre of gravity, according to the rule for finding it given in § 97. So, also, the centre of gravity of an empty box, or an empty ship, would be an imaginary point in the space inside. In a hoop it is the centre of the hoop's circle.

100. =The Centre of Gravity Seeks the Lowest Point.=--The centre of gravity always takes the lowest place which the support of the body will allow. In a suspended body, therefore, it is always directly under the point of suspension. To get to the one side or the other of this position it must rise. This the attraction of gravity forbids, and if by any force it is made to rise, this attraction at once brings it back. This is manifest in the case of a suspended ball, Fig. 39. If the ball be moved to _b_, it will, on being let go, return to its first position, simply because its centre of gravity, in obedience to the earth's attraction, seeks the lowest place possible. From inertia (§ 49) it moves beyond this point, and continues to vibrate back and forth for some time; but when its motion is stopped, it hangs perpendicularly; that is, in such a way that its centre of gravity shall have the lowest possible position. I add a few other illustrations of the same point. When a rocking-horse is at rest, its centre of gravity is directly over the point at which it touches the floor, for thus it has its lowest possible place. If it be rocked, the centre of gravity is moved to a higher point, and for this reason it rocks back again. The same is seen in the swing, the cradle, the rocking-chair, etc. Most interesting illustrations of the same thing are found in the Laggan, or Loggan Stones as they are called, several of which are seen on the rugged parts of the British coast. An immense rock, which has been loosened by some convulsion, rests with a slightly-rounded base on another rock which is flat, and it is so nicely balanced that one person alone can produce a perceptible rocking motion in it. I saw, many years ago, a large rock near Salem, Massachusetts, thus situated. There is one also in Great Barrington, Massachusetts.

101. =Farther Illustrations.=--It is because the centre of gravity always seeks the lowest place that an egg lies upon its side. When on its side, the centre of gravity is at its lowest point, as is manifest from comparing Fig. 40 with Fig. 41 (p. 71). Children often have a toy, called a witch, which illustrates the same thing in another way. It is a piece of light substance, as pith, with a shot fastened in one end. It always stands up on its loaded end, and can not be made to lie down on its side, because the centre of gravity would not then be at the lowest point. There is an amusing Chinese toy of the same kind. It is a figure of a fat old woman, Fig. 42, loaded with lead at the bottom, so that its centre of gravity is at _a_. If the figure be thrust over to one side, as shown by the dotted lines, the centre of gravity is raised, and the upright position is at once resumed. If the toy were not loaded, it would lie in the position represented in Fig. 43, just as the egg lies on its side.

102. =Curious Experiments.=--You can not hang a pail of water on a stick laid upon a table, as represented in Fig. 44, for the centre of gravity is not supported. But if you place another stick a as a brace, in the manner represented in Fig. 45 (p. 73), so as to push the pail under the table, it will hang securely, because the centre of gravity is now under the point of suspension. The explanation of the following experiment is the same: Run a large needle through a cork; fasten to the cork a fork, and you can suspend the whole on the edge of a table, as seen in Fig. 46. Here the centre of gravity is directly under the point of suspension, which is at the point of the needle. The same can be said of the very common toy represented in Fig. 47. The horse, made of very light material, stands securely, because the centre of gravity of the whole is in the heavy ball, which is under the point of suspension. If the horse be made to rock back and forth, the centre of gravity in the ball moves in a curved line, as in the case of a ball suspended by a string (Fig. 39). It is at its lowest place only when the horse is at rest. The hanging of a cane with a hook-shaped handle on the edge of a table is to be explained in the same way.

103. =Stability of Bodies.=--The firmness with which a body stands depends upon two circumstances--the height of its centre of gravity, and the extent of its base. The lower the centre of gravity, and the broader the base, the firmer does the body stand. A cube, represented in Fig. 48, is more stable, that is, less easily turned over, than a body shaped as Fig. 49, because it has a larger base. The contrast is still greater between Figs. 48 and 50. The reason of the stability of a body with a broad base is found in the fact, that in turning it over the centre of gravity must be raised more than in turning over one of a narrower base. The curved lines indicate the paths of the centres of gravity as the bodies are turned over. In the case of a perfectly round ball, the base is a mere point, and therefore the least touch turns it over. Its centre of gravity does not rise at all, but moves in a horizontal line, as shown in Fig. 51. The pyramid is the firmest structure in the world, because it possesses in the highest degree the two elements--a broad base, and a low position of the centre of gravity. On both these accounts the centre of gravity must ascend considerably when the body is turned over, as seen in Fig. 52.

104. =Bodies not Upright Unstable.=--When a body does not stand upright, its stability is diminished simply because only a portion of the base is concerned in its support. In Fig. 53 the base is broad, but the body is so far from being upright that the centre of gravity bears upon the very extremity of the base on one side, as the perpendicular line from it indicates. The least jostle will turn it over, because the centre of gravity need not ascend the least when this is done. You see, then, that the less upright a body is, the less of the base is of service in its support, because the farther is the line of direction of the downward pressure of the centre of gravity from the centre of the base. The famous tower of Pisa, Fig. 54, one hundred and thirty feet high, overhangs its base fifteen feet. It was undoubtedly built intentionally in this way to excite wonder and surprise, for what would otherwise have been a very unsafe structure is rendered stable and safe by the arrangement of its materials. Its lower portion is built of very dense rock, the middle of brick, and the upper of a very light porous stone. In this way the centre of gravity of the whole structure is made to have a very low position.

105. =Familiar Illustrations.=--You see now the explanation of the fact which common experience teaches every one, that the taller is a body, and the narrower its base, the more easily is it overturned. This is exemplified in the two loads, Fig. 55. The base is the space included by the wheels. The centre of gravity is so high in the tall load that a perpendicular line drawn from it falls outside of the base if the cart come upon a considerable lateral inclination of the road. But the smaller load, under the same circumstances, is perfectly secure from overturning. A high carriage is more easily overturned than a low one, for the same reason. A stage, if loaded on its top, is very unsafe on a rough road. Stability is given to articles of furniture by making their bases broad and heavy, as you see in tables supported by a central pillar, candlesticks, lamps, etc. The tall chairs in which children sit at table would be very insecure if their legs were not widely separated at the bottom, thus widening the base of support. In the ladder, so commonly used now in picking fruit, a broad base is furnished between the foot of the ladder and the two standards which are spread out to sustain its top.

106. =Support of the Centre of Gravity in Animals.=--The base of support which quadrupeds have, viz., the space included between their four feet, is quite large; and this is one reason that they walk so soon after birth. A child does well that can walk at the end of ten or twelve months, for the supporting base is quite small compared with that of a quadruped. It consists of the feet and the space between them. It requires skill, therefore, in the child to manage the centre of gravity in standing and walking, and this is gradually acquired. If one should grow up without ever standing on his feet, he would find, as the infant does, that some training is necessary to enable him to do it. It is on account of the smallness of the base furnished by the feet that the statue of a man is always made with a large base or pedestal. Although we exert considerable skill in walking, by no means so much is requisite as the Chinese ladies must put in exercise with their small feet. Still more skill is exercised by one who has two wooden legs, or one who walks on stilts. The base made by the feet can be varied much by their position. If the toes be turned out and the heels brought near to each other, the base will not be as large as when the feet are straight forward and far apart, as is manifest in Figs. 56 and 57. It is for this reason that the child, in his first essays at standing and walking, instinctively manages his feet as in Fig. 56.

107. =Motions of the Centre of Gravity in Walking.=--In walking, the centre of gravity is alternately brought over one foot and the other, and so moves in a waving line. This is very manifest as you see people before you going down the aisle out of a church. When two are walking together, if they keep step the two waving lines of their centres of gravity run parallel, as in Fig. 58, and they walk easily; but if they do not keep step these lines run as in Fig. 59, and the movement is both awkward and embarrassing. The line of movement of the centre of gravity is always slightly waving _upward_ also, as seen in Fig. 60 (p. 78). In the case of a man with wooden legs the line would not be gently waving, but somewhat angular, as represented in Fig. 61.

108. =The Centre of Gravity and Attitudes.=--The object of various attitudes assumed under different circumstances is to keep the centre of gravity over the base of support. A man with a load on his back would not assume the position of Fig. 63, but that of Fig. 62, so that the centre of gravity of his load may be directly over his feet. So a man carrying any thing in front leans backward, as in Fig. 64. In ascending a hill a man appears to lean forward, and in descending to lean backward; but in fact he is in both cases upright in reference to the plain on which the hill stands, as may be seen in Fig. 65. A perpendicular line drawn from his centre of gravity strikes the ground midway between the feet, that is, in the middle of the base, and if prolonged would go straight to the centre of the earth. When one rises from a chair he draws his feet backward, and then bends his body forward to bring the centre of gravity over the feet. Unless this is done, it is impossible to rise, at least deliberately, as you will find by trying the experiment. A man standing with his heels close to a wall can not stoop forward and pick up any thing, for the wall prevents him from moving any part of his body backward, and therefore when he stoops forward, the centre of gravity being brought far in advance of the base, he loses his balance and falls. A man who did not understand this undertook to stoop in this way to pick up a purse containing twenty guineas, which he was to have if he succeeded, the forfeiture in case of failure being ten guineas. Of course his centre of gravity made him lose his wager.

109. =Rope-Dancers, Tops, etc.=--Great skill is exhibited by the rope-dancer in supporting the centre of gravity. Similar skill is seen in feats of balancing, as, for example, in balancing a long stick upright on the finger. In these cases the centre of gravity is very little of the time directly over the point of support. It is kept in constant motion nearly but not quite over this point, this unstable equilibrium, as it is called, being vastly less difficult to maintain than stable equilibrium; that is, keeping the balance in one unvarying position. It is the motion of the top that makes it to stand upright upon its point--a very beautiful example of unstable equilibrium. The centre of gravity revolves around a perpendicular line, at exceedingly little distance from it at first, but greater and greater as its motion becomes less rapid, till at length the centre of gravity gets so far from this line that the top falls. For a similar reason an intoxicated man may not be able to keep himself up if he undertake to stand still, and yet may do so if he keep moving. As in the case of the top, his centre of gravity must be kept in motion, or he must fall.