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Examples of Stable Equilibrium.

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stable, unstable, and neutral equilibrium upon a horizontal plane. The letter g in each shows the position of the centre of gravity.

46. Examples of stable equilibrium.-From what has been said it follows, that the wider the base on which a body rests the greater is its stability; for then, even with a considerable inclination, its centre of gravity is above its base.

The well-known leaning towers of Pisa and Bologna are so much out of the vertical that they seem ready to fall any moment; and yet they have remained for centuries in their present position, because their centres of gravity are above the base. Fig. 31 represents the tower of Bologna, built in the year 1112, and known as the Garisenda. Its height is 165 feet, and it is 7 or 8 feet out of the vertical. The leaning is due to the foundations having given way. The tower on the side is that of Asarelli, the highest in Italy.

In the cases we have hitherto considered the position of the centre of gravity is fixed: this is not the case with men and animals, whose centre of gravity is continually varying with their attitudes, and with the loads they support.

When a man, not carrying anything, stands upright, his centre of gravity is about the middle of the lower part of the pelvis, that is, between the two thigh bones. This, however, is not the case with a man carrying a load, for his own weight being added to that of the load, the common centre of gravity is neither that of the man nor of his burden.

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In this case, in order to retain his stability, the man must so modify his attitude as to keep his centre of gravity above the base

formed by his two feet. Thus a porter with a load on his back is obliged to lean forward (fig. 32), while a man carrying a load in one hand is obliged to lean his body on the opposite side (fig. 33).

Again, it is impossible to stand on one leg if we keep one side of the foot and head close to a vertical wall, because the latter prevents us from throwing the body's centre of gravity vertically above the supporting base.

In the art of rope-dancing the difficulty consists in maintaining

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the centre of gravity exactly above the rope.

In order more easily

to accomplish this the performer holds in his hands a long pole,

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The Balance.

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which, as soon as he feels himself leaning on one side he inclines towards the opposite one; and thus contrives to keep the centre of gravity common to himself and the pole, above the rope, and so preserves his equilibrium.

47. The balance.-The balance is an instrument for determining the relative weights or masses of bodies. There are many varieties.

The ordinary balance (fig. 34) consists of a lever of the first kind, called the beam, with its fulcrum in the middle; at the extremities of the beam are suspended two scale pans, D and C; one intended to receive the object to be weighed, and the other the counterpoise. The fulcrum consists of a steel prism, n, commonly called a knife edge, which passes through the beam, and rests with its sharp edge, or axis of suspension, upon two supports; these are formed of agate, or polished steel, in order to diminish the friction. A needle or pointer is fixed to the beam, and oscillates with it in front of a graduated arc, a; when the beam is perfectly horizontal the needle points to the zero of the graduated arc.

Since (33) two equal forces in a lever of the first kind cannot be in equilibrium unless their leverages are equal, the length of the arms A and B ought to remain equal during the process of weighing. To secure this the scales are suspended from hooks, whose curved parts have sharp edges, and rest on similar edges at the ends of the beam. In this manner the scales are supported on mere points, which remain unmoved during the oscillations of the beam. This mode of suspension is represented in the above figure.

The weight of any body is determined by placing it in one of the pans of the balance D, for instance, and adding weights to the other until equilibrium is established, which is the case when the beam is quite horizontal.

48. Conditions to be satisfied by a good balance.—A good balance should be accurate, that is, it should give exactly the weight of a body; it should also be delicate, that is, the beam should be inclined by a very small difference between the weights in the two scales.

Conditions of accuracy. i. The two arms of the beam ought to be precisely equal, otherwise, according to the principle of the lever (33), unequal weights will be required to produce equilibrium. To test whether the arms of the beam are equal, weights are placed in the two scales until the beam becomes horizontal; the contents of

the scales being then interchanged, the beam will remain horizontal if its arms are equal, but if not, it will descend on the side of the longer arm.

ii. The balance ought to be in equilibrium when the scales are empty, for otherwise unequal weights must be placed in the scales in order to produce equilibrium. It must be borne in mind, however, that the arms are not necessarily equal, even if the beam remains horizontal when the scales are empty; for this result might also be produced by giving to the longer arm the lighter scale.

iii. The beam being horizontal, its centre of gravity ought to be in the same vertical line with the edge of the fulcrum, and a little below the latter. For if the centre of gravity coincided with this line, the action of gravity on the beam would be null, and it would not oscillate. If the centre of gravity were above the edge of the fulcrum the beam would be in unstable equilibrium; while, if it is below the fulcrum, the weight of the beam is continually tending to bring it back to the horizontal position as soon as it diverges from it, and the balance oscillates with regularity.

Conditions of delicacy. 1. The centre of gravity of the beam should be very near the knife edge; for then, when the beam is inclined, its weight only acting upon a short arm of the lever, offers but little resistance to the excess of weight in one of the pans.

2. The beam should be light; for then the friction of the knife edge upon the supports is smaller the less the pressure. In order more effectually to diminish friction, the edges from which the beam and scales are suspended are made as sharp as possible, and the supports on which they rest are very hard.

3. Lastly, the longer the beam the more delicate is the balance; because the difference in the weights in the pans then acts upon a longer arm of the lever.

49. Method of double weighing.—Notwithstanding the inaccuracy of a balance, the true weight of a body may always be determined by it. To do so, the body to be weighed is placed in one scale, and shot or sand poured into the other until equilibrium is produced; the body is then replaced by known weights until equilibrium is re-established. The sum of these weights will necessarily be equal to the weight of the body, for, acting under precisely the same circumstances, both have produced precisely the same effect.

50. Weighing machines.-One of the forms of these instru

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Weighing Machines.

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ments, which are of frequent use in railway stations, coal yards, etc., for weighing heavy loads, is represented in fig. 35. It consists of a platform, A, on which the body to be weighed is placed,

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and to which an upright B is fixed; the whole rests on a frame, HE, by the following mode of suspension.

To the upright, E, are adapted two pieces of iron, which support a beam, LR, by the aid of a knife edge, which traverses it at O.

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The two arms of the beam are unequal in length; one of them supports a scale, D, in which are placed the weights; the other arm

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