Page:Elementary Text-book of Physics (Anthony, 1897).djvu/92

 '''63. The Balance.'—The weights of bodies, and hence also their masses, are compared by means of the balance''.

To be of value, the balance must be accurate and sensitive; that is, it must be in the position of equilibrium when the scale-pans contain equal masses, and it must move out of that position on the addition to the mass in one pan of a very small fraction of the original load.

The balance consists essentially of a regularly formed beam, poised at the middle point of its length upon knife-edges which rest on agate planes. From each end of the beam is hung a scale-pan, in which the masses to be compared are placed. Let $$O$$ (Fig. 27) be the point of suspension of the beam; $$A, B$$, the points of suspension of the scale-pans; $$C$$, the centre of gravity of the beam, the weight of which is $$W$$. Represent $$OA = OB$$ by $$l$$, $$OC$$ by $$d$$, and the angle $$OAB$$ by $$\alpha$$.

If the weight in the scale-pan at $$A$$ be $$P$$, and that in the one at $$B$$ be $$P + p$$, where $$p$$ is a small additional weight, the beam will turn out of its original horizontal position, and assume a new one. Let the angle $$COC'$$, through which it turns, be designated by $$\beta$$. Then the moments of force about are equal; that is, from which we obtain, by expanding and transposing,  The conditions of greatest sensitiveness are readily deducible from this equation. So long as $$\cos{\alpha}$$ is less than unity, it is evident that $$\tan{\beta}$$, and therefore $$\beta$$, decreases as the weight $$2P$$ of