344 



THE POPULAR EDUCATOR. 



13 12 II 10 8*K 65132 I 



it is essential that its own centre of gravity should lie some- 

 where on the short arm ; for then the counterpoise can balance 

 it when placed in some position on the other arm, such as that 

 marked o, in the figure. For this reason steelyards are made 

 heavy at one end. 



To Graduate a Steelyard. The centre of gravity of the beam 

 being on the hook side of the fulcrum, let it be brought into an 



horizontal position, 

 no weight being on 

 the hook. Then, as 

 proved in Lessons 

 VH. and VIII., the 

 moment of P is equal 

 to the moment of the 

 beam, that is, the 

 weight of the beam 

 multiplied into the 

 distance of its centre 

 of gravity from a ver- 

 tical line through the 

 Fig. 54. fulcrum, is equal to 



P multiplied into the 



distance of o from that line. At the point o so found draw a 

 line across the beam ; that line represents the zero division of 

 the long arm, or the division at which P produces equilibrium, 

 the weight on the hook being nothing, cipher, or zero. 



Now, supposing that any number of pounds, w, of any sub- 

 stance are hung on the hook, while P is shifted to the left until, 

 as in the figure, the arm is again horizontal, we have P multi- 

 plied by the distance of its ring from the fulcrum a equal to w 

 multiplied by a 6 (this line a b being supposed horizontal), together 

 with the moment of the beam. But P multiplied by the distance 

 of the zero division from a, is equal to the moment of the beam, 

 as already proved ; therefore it follows that P multiplied by its 

 distance from the zero division is equal to w multiplied "by a b. 

 Now, in order to graduate, let us suppose P one pound and w 

 seven. Then we have in numbers seven times a, b equal once 

 the distance of the counterpoise from o, which tells us the exact 

 position of P for 7 pounds on the hook, namely, that you find it by 

 measuring from o to the left seven pieces each equal to a b. 

 Let w be 13 pounds or 3 pounds, then in like manner you 

 measure 13 or 3 pieces equal to a b. It thus appears that the 

 subdivisions for the successive pounds are equal to each other ; 

 and we may therefore lay down the following rule for graduating 

 a steelyard : 



Find first the zero subdivision by bringing the unloaded in- 

 strument into an horizontal position by the counterpoise. Put 

 then on the hook, or in the pan, such a number of even pounds 

 aa will push the counterpoise to the greatest distance it can go 

 on its arm for even pounds, and divide the distance between this 

 last position and the zero point into as many equal parts as 

 there are then pounds on the hook. The points of division so 

 obtained are the positions of the counterpoise for the several 

 pounds up to that number. 



For half and quarter pounds these divisions must be sub- 

 divided ; and for greater weights than one pound will balance 

 on the long arm, the counterpoise must be doubled or trebled, 

 etc. If the steelyard be intended for weighing small objects, 

 such as letters, the counterpoise may be ounces, or tenths 

 of an ounce, or even smaller weights, as occasion requires. 

 It thus appears that the construction of a steelyard is very 

 simple, and that any handy person of a mechanical turn 

 may make one of steel or iron, 

 or even of a piece of hard 

 wood, without much trouble. 



THE DANISH BALANCE. 



This is a species of steel- 

 yard, in which (Fig. 55) the 

 fulcrum is movable, and the 

 counterpoise is the weight of 

 the beam acting at its centre 

 of gravity, P, the substance to be weighed being suspended 

 from a hook or placed in a pan, at the extremity, R, on 

 the other side of the fulcrum. The question is, how may you 

 graduate such an instrument ? To do this, let us suppose 

 the beam to weigh 1 pound, and that 1 ounce of some substance 

 is placed in the scale ; then it is evident that the fulcrum, F, 



Fig. 55. 



must be shifted to the point in which F R is to F P in the pro- 

 portion of 16 to 1, there being 16 ounces in the pound. This 

 comes to dividing the distance R p (which is known) into 

 seventeen equal parts, as proved in Lesson IV., and taking the 

 first point of division next to p for the fulcrum. If there be 2 

 ounces in the pan, RF must be to F p as 16 to 2 ; that is, you 

 divide R P into 18 parts, and take the fulcrum 2 from p. If 

 there be 7 ounces, you divide into 23 parts, and take 7 next to 

 P; and so on for all the ounces from 1 to 16 you may determine 

 the several positions of the fulcrum, marking them as you pro- 

 ceed. If the beam be of any other weight, you follow a similar 

 course, dividing R P into as many equal parts as there are ounces 

 in the sum of the weights of the beam and substance, and count- 

 ing off as many divisions from P as there are ounces in the 

 latter. 



From all this it is evident, first, that the subdivisions are not 

 equal to each other, as in the steelyard; secondly, that the 

 operation of graduation is more troublesome than in that instru- 

 ment. The Danish balance, however, has the advantage of not 

 being encumbered with a movable counterpoise ; it carries its 

 own imperial standard weight within itself. 



THE BENT LEVER BALANCE. 



The principle of this instrument, a species of which is largely 

 sold for weighing letters, may be understood by the aid of the 

 accompanying Fig. 56. On an upright stand is placed a quadrant 

 arc, M o, of which c is the centre. Round c as a fulcrum revolves 

 a lever, usually bent, but in the figure represented as formed of 

 two arms at right angles to each other. The arm c B is gene- 

 rally of small weight, being lightly constructed, while the other, 

 CG, called the "index arm," is heavily weighted at its lower 

 end, the centre of gravity of the whole lever thus being nearly 

 ^ c at some point, G, on that arm. 



On some substance, w, to be 

 weighed, being suspended 

 from B, the index moves from 

 its zero point, o, up the quad- 

 rant until the weight of the 

 lever acting at G balances w 

 at B, that is, until the mo- 

 ments of these forces are 

 equal, which will be when w 

 multiplied by B H is equal to 

 the weight of the lever mul- 

 tiplied by G I. The divisions 

 of the quadrant corresponding 

 to the several weights 1, 2, 3, 4> etc., suspended from B are, 

 however, best determined by experiment for each weight. 



THE LEVER WHEN THE FORCES ARE NOT PARALLEL. 



In all the cases of levers and weighing instruments wo have 

 so far considered, the forces were supposed parallel in weighing 

 instruments necessarily so. The treatment of the subject is, 

 however, not complete until the condition of equilibrium is de- 

 termined for levers the forces acting on which are not parallel. 

 This is the most general case that can occur, and indeed it 

 includes all the others. To clearly understand it, let a lever be 

 defined a mass of matter of any shape ivhich has one fixed point 

 in it. It may be a bar straight, or simply bent, or bent and 

 twisted, or it may be a solid block. So long as there is one 

 point fixed, we may treat it as a lever, that point being the 

 fulcrum. 



Moreover, the two forces which act on it are supposed to be 

 such that their directions when produced meet, and that their 

 plane passes through the fulcrum. In cases where the two 

 forces do not meet, or their plane does not pass through the 

 fulcrum, there cannot be equilibrium. For example, the out- 

 stretched right arm of a man is a lever, of which the fulcrum is 

 in the right shoulder. Suppose, as he stretches it before him in 

 a horizontal position, one force is applied to the hand obliquely 

 from him towards the left to the ground, while another acts 

 horizontally at his elbow towards the right and at right angles 

 to the arm ; these forces cannot meet, and therefore would not 

 under any circumstances keep the arm in equilibrium ; further, 

 even were they to meet, they would not so keep it unless their 

 plane passed through the fulcrum in the shoulder socket. Sup- 

 posing the forces, therefore, to be as described, namely, that 

 their directions meet and their plane passes through the fulcrum, 



Fig. 56. 



