78

STATICS.

difference of P and Q is: hence

tan 8 ,

-is a correct measure

P-Q.

of the sensibility : and therefore the second requisite is fulfilledh k

by making (P + Q) - + W - as small as possible.

The stability is greater the greater the moment of theforces which tend to restore the equilibrium when it is de-stroyed. Suppose P = Q, then P and Q may be placed at themid-point between A and B : and the moment of the forcestending to restore equilibrium equals j(P + Q)h+ Wk^ sin 8.Hence to satisfy the third requisite, this must be made as largeas possible. This is, in part, at variance with the second re-quisite. They may, however, both be satisfied by making(P + Q) h + Wk large, and a large also: that is, by increasingthe distances of the fulcrum from the beam and from the centreof gravity of the beam and scales, and by lengthening thearms.

It must be remarked that the sensibility of a balance isof more importance than the stability, since the eye can judgepretty accurately whether the index of the beam makes equaloscillations on each side of the vertical line; that is, whetherthe position of rest would be horizontal: if this be not thecase, then the weights must be altered till the oscillations arenearly equal.

97- Another kind of balance is that in which the armsare unequal, and the same weight is used to weigh differentsubstances by varying its point of support, and observing itsdistance from the fulcrum by means of a graduated scale.The common steelyard is of this description.

Prop. To shew how to graduate the common steelyard.

98. Let AB be the beam of the steelyard (fig. 34 .) Athe fixed point from which the substance to be weighed issuspended, Q being its weight: C the fulcrum : W the weightof the beam together with the hook or scale-pan suspendedfrom A ; G the centre of gravity of these.

Suppose that P suspended at N balances Q suspended fromA, then taking the moments of P, Q, W about C, we have