« PreviousContinue »
stands securely, and even when one wheel is raised a little by passing over a stone, it rights itself again as soon as possible. But let the stone be a large one, or let the wheel rise on a high bank, and the cart will probably be upset. The bodies of living animals are aided by their muscular activity in maintaining any desired posture, yet there are many postures which we ourselves cannot assume without the risk, or even the certainty of a fall.
It thus becomes an interesting question, why a body can stand only in certain positions, and what are the conditions of its standing or falling. In order to answer this, we must consider once more the nature of the force of gravity, which produces these effects. It must be borne in mind that this force acts on every particle of a body independently of the rest. Every particle is urged, as if by a separate impulse, towards the centre of the earth. We have therefore to consider, not a single force, but a number of separate forces acting in parallel lines and in the same direction. Now there is in every body, a certain point round which all these forces balance each other. If this point be fixed, the forces on any one side of it will tend to pull that side downwards, which they can do only by raising the other side; the forces on the other side have an exactly opposite tendency; and, since they balance each other, no motion will ensue. In short, there is a certain point in every body, such that, if it be fixed, gravity cannot make the body move. The whole force of gravity, then, on the different parts of the body, may be considered as collected at that point, for it is only by moving that point that gravity can produce any motion at all. The point in question is therefore called the centre of gravity. Every body has a centre of gravity, though in some its position is not so easily ascertained as in others. In a regular symmetrical body, it is quite easy to see where it must be. For instance, in a straight rod of uniform thickness and density, it must be at an equal distance from both ends. In a sphere, or a spheroidal body like the earth, it is at the centre. Its position may be determined by mathematical reasonings, in all
bodies of regular figure; in others it has to be found by experiment.
It will now be easy to understand why a body remains at rest in some positions, and not in others. The whole force of gravity upon it, or, in other words, its whole weight, may be considered as one single force acting downwards from its centre of gravity, and urging that point to descend. If the centre of gravity is free to obey the impulse, it will descend accordingly, but if it be supported, the whole body will remain at rest.
Suppose a body E (fig. 25), whose centre of gravity is at G, to stand slantingly on a horizontal surface. The weight of the body acts in the vertical line G I, which is called the line of direction. Since this line falls within the base of the body E, the centre of gravity cannot move downwards, and the body will stand. But if a heavy body F be placed on the top of the body E, the common centre of gravity of the two bodies will evidently be at some point above G. Let it be at H. The line of direction is now H K, which falls beyond the base on which the bodies rest. The centre of gravity is no longer supported, and both bodies will tumble down. It appears from this example, that a body stands the more securely, the lower the position of its centre of gravity. Every one knows how dangerous it is to load a cart, a coach, or (worst of all) a boat, in such a way as to make it topheavy. The base on which a coach rests is the space inclosed by the wheels; so that, if the line of direction fall beyond the wheels, the coach will certainly be overturned. The higher the centre of gravity, the more likely is this disaster to occur. When a boat, in a stormy sea, is almost capsized by the heaving of the waves, the passengers, in their alarm, are apt to start up from their seats. Nothing can be more foolish; for, by their doing so, the centre of gravity is raised, and the danger consequently increased.
The centre of gravity of the human body is situated in the lower part of the trunk. When a man stands, his base is the space covered by the soles of his feet, including also the space, if any, which intervenes between them. porter, carrying a load on his back, leans forward, so that the common centre of gravity of himself and his load may be directly above that base. If he tried to stand erect, the line of direction would pass behind his heels, and he would inevitably fall backwards. A nurse with a child in her arms leans back for a like reason. A sailor acquires the habit of walking with his feet far apart, thus giving himself a broader base, that he may tread more steadily on the moving deck.
It is unnecessary to multiply examples; the thoughtful reader will find them for himself at every turn. He should now be able to tell, without further explanation, why a man, with a pitcher in one hand, leans to the other sidewhy a person stooping forward advances one foot-why a very fat man carries his head and shoulders so far backwhy a man standing on one foot inclines his body to the same side—why, in walking, we move ourselves alternately from one side to the other-why two persons, walking arm in arm, jostle each other when they do not keep the stepwhy a person in danger of falling stretches out his arm, and perhaps his leg, in the opposite direction-and finally, why a person cannot rise from his seat without either bending his body forward, or drawing his feet backward. When all this has been duly considered and understood, it can hardly fail to suggest the reflection, how little we are conscious of the profound and far-reaching principles, which often regulate the most trivial actions of our daily life.
CENTRIFUGAL AND CENTRIPETAL FORCE.
THE earth attracts a stone, and the stone falls; the earth attracts the moon, but the moon does not fall. Why is this? Not because the moon is larger than the stone, for
the larger it is, the greater the force with which it is attracted. Nor yet because the moon is farther away than the stone, for though the distance does greatly diminish the force of the earth's attraction, it cannot destroy it altogether. There can be no doubt that the moon would move towards the earth, slowly at first, perhaps, but not the less surely, if the attraction of the earth were not counteracted by some other force. Now what is that force?
The best way to arrive at the answer to this question, will be to consider, first, the motion of a stone whirled round the hand in a sling. Every boy knows how tightly it stretches the cords of the string, as if it wished to get away. It requires a constant muscular effort of the hand to keep it in a circular, or nearly circular course. There seems to be some force impelling it outwards, and whenever one of the cords is let go, it accordingly flies off with great velocity. Now, the point to be determined is this:-What is the force which gives it this outward impulse? The hand is the centre round which the stone revolves in something like a circle, and towards which the cords of the sling, or rather the hand acting through them, is constantly pulling it. But this force which the hand exerts, and which is called, for an obvious reason, a centripetal (or centre-seeking) force, does not succeed in actually bringing the stone to the centre round which it moves. Why does it not? What is that counteracting influence, which urges the stone away from the centre of revolution, and may therefore be called, as it has usually been, a centrifugal (or centre-flying) force?
It is surprising to find that this centrifugal force is nothing else than inertia. Suppose the hand to be at A (fig. 26), and the stone to be whirled round it in the direction shown
by the arrows. When the stone has reached the point B, it is moving, for the instant, in the direction of the straight line BE, which touches the circle at that point, and is therefore called a tangent. If every external force now ceased to act upon it, its inertia would carry it, without any change of velocity, along the tangent towards E. And this is nearly what happens when one of the cords of the
sling is let go, and the stone thereby set free. But so long as it is confined in the sling, it is clear that it cannot proceed towards E, unless the hand were to move from A. It accordingly pursues a course as near the line BE, as the length of the sling will permit; that is to say, it stretches the cords in that direction, and describes the curve BD. Its circumstances are similar in many respects to those of a projectile. Impelled by its inertia towards E, and pulled by the cords of the sling towards A, it takes a middle course between these two directions, and, as in the case of projectiles, that course is curvilinear. When it arrives at D, its tendency to move along the tangent DG is counteracted and modified as before, by the cords continuing to pull it towards A.
It is this tendency to "fly off at a tangent," as it is often expressed, to which the name of centrifugal force has been given. It is to be observed that it is not a force acting directly away from the centre round which a body revolves. But it urges the revolving body in a direction which would carry it every moment farther and farther from that centre, so that the term "centrifugal" cannot be pronounced inaccurate.
In considering the motion of the stone in a sling, no account has been taken of its weight, that is, of the effect of gravity upon it. But it is worth while to notice, that the stone does not fall, even when it is at the top of its circuit, and has consequently no support beneath it. The reason is, that the centrifugal force is greater than the weight of the stone, and therefore keeps it in its place. The same thing may be shown still more strikingly by whirling a pail of water in the hand. It will be found that, if the pail be whirled rapidly enough, though the mouth of it be presented downwards at one part of each revolution, not a drop of the water will be spilt.