Chap. XT.
MECHANICS.
929
In a lever of the second kind, the weight is ^placed between the power and the fulcrum, asin cutting instruments which move round the t.
point of the blade, &c.
'i _
R
C
P : W=BC : AC.
In a lever of the third kind, the power isplaced between the weight and the fulcrum, asin fire-tongs, &c.
P : \V=BC : AC;
Fig. 1252
WxBC
Px AC
hence we have P=
WxBC
Px AC
Fig. 1253.
The weight supported by the fulcrum C is equal to the sum of the power and weight, orp + \v, when they act on different sides of the fulcrum, and therefore in the same direction ;and to the difference, P — W, when they act on the same side of the fulcrum, and thereforein different directions. If the power P is shifted along the arm C A of the lever A B ofthe first class, the weight W, which is capable of maintaining an equilibrium, will be pro-portionate to CA , the distance of the power P from the fulcrum; for since P : W = BC: A C, we have W : A C = P : B C; and if a be another value of W, and a C another valueof AC, we shall then have w : a C = P; BC. In a lever of the second kind the powermust be less than the weight, since it must be further from the fulcrum; and in one of thethird kind it must be greater than the weight, as it is nearer to the fulcrum. When thecentre of gravity of the lever or bar is not immediately over the fulcrum, the weight of thebar must be taken into account. Of levers of the first kind there are many examples, asthe crowbar, scissors, snuffers, &c., where two levers are riveted together : in elevatinga ladder the first process partakes of a lever of the second kind, and afterwards of thethird. When a man raises a stone by the application of the crowbar resting on a fulcrumbetween himself and the stone, he presses down the end of the lever, and the utmost forcehe can apply is equal to the whole weight of the body; but when he thrusts the
lever under the stone so that its extremity bears on the ground, it becomes a lever of thesecond class; and in order to raise the stone he must draw the end of the lever upwards,
and in this direction a powerful man can exert a force equal to twice his weight.
The Roman Steelyard is usually ___
formed of a bar of iron resting upon
an edge or pivot, and having one arm j~ | _ ** __ —j _ A n
longer than the other. To graduate c I i ^ I -II 3
this machine or the arm C B : let us i c* X.
suppose M to be the weight of it when j p |
united to its centre of gravity ti, N the p. ■ *
weight of C A united in its centre of
gravity v, and O the weight of the scale: let P, P', P", be the different weights employed :an equilibrium is obtained by placing the constant weight, W, at the different distancesa,b,c; consequently, there will be an equilibrium in the following cases: —
Px CA + N x Ctt-j-O x C A = W x C a -f M x Cm;P'xCA + NxCra-rOxCA = Wx C6+MxCm;P"x CA + NxCn+Ox CA=WxCc + Mx Cm.
By subtracting the first equation from the second, and the second from the third, we have
P'-PxCA
If the weights are in arithmetical progression, the distances a 6,6 c, will be equal, or form