To exemplify the principle of the lever, let the bar A B (Fig. 272) be balanced accurately with the scale platform, but without the weights R and P. Then, placing the article R upon the platform, move the weight P along the beam until there is an equilibrium. Suppose the distances A C and B C are found to be 2 and 40 inches respectively, and suppose the weight P to equal 5 pounds, what at this point will be the weight of R? By trial we shall find that R = 100 pounds. Again, if a portion of R be removed, then the weight P would have to be moved along the bar B C to produce an equilibrium; suppose it be moved until its distance from C be found to be 20 inches, then the weight of R would be found to be 50 pounds, or -
R = 50 pounds.
Again, suppose a part of the weight taken from R be restored, and the weight P, on being moved to a point required for equilibrium, be found to measure 30 inches from C, then we shall find that -
R = 75 pounds.
Thus when -
BC = 40, R = 100; or, 100/100 = 2 . 5;
BC=30, R = 75; or, 75/30 = 2.5;
BC = 20, R=50; or, 50/20. =2.5;
If, instead of moving P along B C, its position be permanent, and the weight P be reduced as needed to produce equilibrium with the various articles, R, which in turn may be put upon the scale; then we shall find that if when the weight P equals 5 pounds the article R equals 100, and there is an equilibrium, then when -
P= 9/10x 5 =4. 5, R will equal 9/10 x 100 = 90;
P = 8/10 x 5 = 4, R will equal8/10 x 100 = 80; P= 7/10 x 5 = 3. 5, R will equal7/10 x 100 = 70;
and so on for other proportions; and in every case we shall have the ratio R/P equal 20, thus -
R/P = 90/4.5 = 20 •
R/P = 80/4 = 20;
R/P = 70/3.5 = 20;
Thus we have an equality of ratios in comparing the weights.
Again, if the weight P and the article R be permanent in weight, and the distances A C, B C be made to vary, then if there be an equilibrium when A C is 2 and BC is 40, we shall find that when -
A C = 8/10 x2=1.6; B C will equal 8/10 x 40 = 32 ,
A C = 6/10 x 2 = 1 . 2; B C will equal 6/10 x 40 = 24;
AC= 4/10 x 2 = 0.8; BC will equal 4/10 x 40 =16; and so on for other proportions, and in every case we shall have the ratio BC/AC = 20; thus -
BC/AC = 32.1.6 = 20;
BC/AC = 24/1.2 = 20;
BC/AC = 16/0.8 = 20;
producing thus an equality of ratios in comparing the arms of the lever. From these experiments we have found, in comparing the article weighed with an arm of the lever, the constant ratio B C: R, and when comparing the weights we have found the constant ratio P: R. Again, in comparing the arms of the lever, we find the constant ratio A C: B C. Putting two of these couples in proportion, we have -
A C: B C:: P: R. Hence (Art. 373) we have -
ACxR = BCxP.
Dividing both members by A C, we have -
In a steelyard the short arm, A C, and the weight, or poise, P, are unvarying; therefore we have -
R = BCxP/AC;
or, when P/AC is constant, we have -
R: B C.