of the term moment we may express the relation which must hold between the Power and the Weight for equilibrium on the Lever thus: the moments of the Power and the Weight with respect to the fulcrum must be equal. It is easy to see that this coincides with what is stated in Art. 206. 211. In the present Chapter we have left out of consideration the fact that the rod or bar of the Lever will itself have weight. If the fulcrum be at the centre of gravity of this rod or bar the weight of the rod or bar is entirely supported by the fulcrum, and so need not be regarded; the rod or bar so far as we are concerned with it is practically without weight. But if the fulcrum is not at the centre of gravity of the rod or bar, allowance must be made for the weight of the rod or bar: the account of the Common Steel-yard in the next Chapter will illustrate this point. XIII. THE BALANCE. 212. The various kinds of Balances form such a very important application of the Principle of the Lever that we shall devote a separate Chapter to them. The use of the Balance, as is well known, is to determine the weight of any proposed body, so that in this case we employ the Lever not to produce motion, but to prevent motion, that is to preserve equilibrium. 213. The Common Balance. The Common Balance consists of a beam with a scale suspended from each end; the beam can turn about a fulcrum which is above the centre of gravity of the beam, so that if the scales were removed the beam would adjust itself to a position of stable equilibrium: see Art. 183. The arms of the beam should be of equal length, and the scales of equal weight, so that the beain may be at rest in a horizontal position when the scales are attached and are empty. If these conditions are satisfied the Balance is said to be true; if not it is said to le false. The body to be T. P. 6 weighed is placed in one scale and weights in the other until the beam remains at rest in a horizontal position. In this case if the Balance be true the weight of the body is indicated by the weights which have been put in the other scale. We may test whether the Balance is true by observing whether the beam still remains at rest in a horizontal position when the contents of the two scales are interchanged. But even if a Balance be false we may determine by its aid the exact weight of a body, if we employ the process which is called double weighing. Put the body which is to be weighed in one scale, and in the other scale put sand or shot so as exactly to counterpoise the body. Remove the body and put in its place weights so as just to restore equilibrium again. Then the sum of these weights indicates the weight of the body. This process of double weighing is very simple in theory and very exact in practice. 214. Another kind of Balance is that in which the arms are unequal, and the same Weight is used to weigh different substances by putting it at different distances from the fulcrum. The Common Steel-yard is of this kind. Let AB be the beam of the Steel-yard, C the fulcrum. Let A be the fixed point from which the body to be weighed is suspended. Let Q be the weight of the beam together with the hook or scale-pan at A. Let P be a weight which may be placed at any distance from the fulcrum. We have now to graduate the Steel-yard, that is to put marks on it so that if we observe the position which P has when a body is suspended from A, and the whole is in equilibrium, we may know the weight of that body. Now we might proceed by the aid of theory. For the weights P and Q being parallel forces we can determine their resultant by Art. 165; and then this resultant must balance the weight of the body, according to the Principle of the Lever. But it will be more simple to proceed by the aid of experiment. Take then a weight, say of one pound, and suspend it from A; move P about until such a place is found for it that the beam just remains in equilibrium, and mark the place with the figure 1. Again instead of the weight of one pound at A put a weight of two pounds; move P about as before, and mark with the figure 2 the place which it has when the beam is in equilibrium. Proceeding in this way the beam becomes graduated, and the Steel-yard is fit for use. It will be found by trial that the figures 1, 2, 3, 4, ... succeed at equal distances on the beam. Thus when we have a body to be weighed we suspend it from A, and then move P about until it comes to such a place that the beam remains at rest in a horizontal position. Let this, for example, be when P is midway between the figures 3 and 4, as in the diagram; then we infer that the body weighs 3 pounds. 216. Sometimes two different graduations are recorded on the Steel-yard, corresponding to two different moveable Weights. In this way we can extend the range of the machine without making the machine itself inconveniently long; thus one graduation might give us the weights of bodies up to 10 pounds, and then another graduation corresponding to a heavier moveable weight might give us the weights of bodies of 10 pounds and upwards to about 100 pounds. Or the two graduations may correspond to two different positions of the point A from which the body to be weighed is hung, the moveable weight P being the same in both cases. 217. Another kind of Steel-yard is called the Danish Steel-yard, This consists of a heavy beam which terminates in a knob at one end; and the body to be weighed is placed at the other end, the fulcrum being moveable. Let AB be the beam; let P denote its weight, and G its centre of gravity. The body to be weighed is suspended from A, and the fulcrum is moved about until there is equilibrium, when the beam is horizontal. The Danish Steel-yard might be graduated by the aid of theory; for P at G must balance the body hung from A according to the Principle of the Lever. Or we may proceed by experiment as before. Let a body of one pound weight be suspended at 4; move the fulcrum about until there is equilibrium with the beam horizontal, and mark the position of the fulcrum by the figure 1. Again, instead of the weight of one pound at A put a weight of two pounds; move the fulcrum about as before, and mark with the figure 2 the place which it has when the beam is horizontal and in equilibrium. Proceeding in this way the beam becomes graduated and the Steelyard is fit for use. It will be found in this case that the figures 1, 2, 3, 4, do not succeed at equal intervals on the bean. Thus in using the Danish Steel-yard to weigh a body if the fulcrum comes precisely under one of the figures marked on the beam we know the weight of the body; but if the fulcrum comes between two of the figures we cannot tell the weight exactly, but only two values between which it must lie. ... 218. There are some weighing machines which do not depend on the Principle of the Lever. They usually consist mainly of a strong spring which is drawn out to a greater extent the heavier the body is which is suspended from it; and a contrivance is furnished by which we can readily observe how far the spring has been drawn out. These machines may be graduated by experiment, that is by suspending known weights and recording the corresponding points to which the spring is drawn out. XIV. THE WHEEL AND AXLE. THE TOOTHED WHEEL. 219. In this Chapter we shall consider two other Mechanical Powers, namely, the Wheel and Axle, and the Toothed Wheel. 220. The Wheel and Axle. This machine consists of W two cylinders which have a common axis; the larger cylinder is called the Wheel and the smaller the Axle. The two cylinders are rigidly connected with the common axis, which is supported in a horizontal position, so that the machine can turn round it. The Weight acts by a string which is fastened to the Axle and coiled round it; the Power acts by a string which is fastened to the Wheel and coiled round it. The Weight and the Power tend to turn the machine round the axis in opposite directions. 221. When there is equilibrium on the Wheel and Axle the Power must be to the Weight in the same proportion as the radius of the Axle is to the radius of the Wheel. For it is easy to see the close resemblance between this machine and a Lever of the first class. It will be obvious that the effect of the Weight must be the same whether it is placed |