MECHANICAL PHILOSOPHY. STATICS. 



[liEST LEVERS. 



LfTrr of Ou third kind. In the l.-vcr of the third kind 

 the power P (Fig. 94) and the weight W act aa in the 

 second kind, in oppo- n,. M. 



into direction! on the 

 aide of the ful- 

 ; but in the 

 third species of lever, 

 the power U nearer 

 the fulcrum than the 

 might 



Moat muscles in 

 animal* are gene- 

 rally inserted near 

 the joint, aa in Fig. 

 05, and act M leven 

 of tho third kind ; 

 the joint forms tho 

 fulcrum ; the limb 

 which the muscle 

 moves, together with 

 any resistance opposed to its motion, represents the 

 fit. 94. 



_ 



r 



weight ; while the force exerted by the contraction of the 

 muscle is the power. The treadle of a turning-lathe, or 

 razor-grinder's machine, where the foot of the operator 

 act* as the power, is a lever of the third kind. 



A pair of tongs, and a pair of sheep-shears, afford 

 familiar instances of double levers of the third kind. 



PROPOSITION XXIII. 

 To find the Conditions of Equilibrium and the Pretsure on 



the Fulerum when two Parallel Forces act in the tame 



plane on a straight lever. 



Let P and W be two parallel forces acting at A and B, 

 on the straight lever A B whose fulcrum is F in each of 

 tho three kinds of levers. 



And let A P and W B (Fig. 9C 1, 2, and 3) represent 

 these forces in Fif. M. 



magnitude and 

 direction. 



In order that 

 there may be 

 equilibrium, tho 

 resultant of the 

 two parallel for- 

 ce* must pass 

 through F, and 

 the pressure on 

 that point will be 

 equal to their re- 

 sultant, and thin 

 Kill be destroyed 



5 the resistance 

 the fulcrum. 

 The proposition 

 U therefore re- 

 duced to the case 

 of Prop. XI., 

 which we hare 

 already demon- 

 strated ; and from 

 this we may learn 

 that the pressure 

 ill be equal 

 to P -f W acting 

 in a direction pa- 

 rallel to P or W. 



Aud also that P AF - W -BF or ~ _ B Jin every 



From this we find that in tho lever of tho first kind, 

 the power maybe less than, equal to, or greata than, 

 the weight ; in tho second kind the ] ower is always less ; 

 ami iu tin- third kind always greater than the Wright. 



In the preceding cases we have su]>i".sid tin- ],-v, r to 

 bo without weight ; we shall show how to t:ik<> tl,.- \\ . ijjit 

 of lever into consideration, in the cases of the balance 

 and steelyard. 



PROPOSITION XXIV. 



To find the Conditions of Equilibrium nf <iro Forces 

 acting in the tame plant on a Unit Lever, when the 

 directions of the Forcet are not Parallel. 



Let ACB (Fig. 97) be a bent lever consisting of a 

 portion of a plane rigid body, Mi]p]>o.-e.il t.i In- destitute 

 of weight. C the fulcrum about which tho k-v.-r turns. 



A and B the points of application of the tno forces 



represented in magnitude and direction l>y A P and 

 BW. 



Tie effect of these forces will be to produce a 

 pressure on C ; and when there is equilibrium, this 

 pressure must be destroyed by the reaction K of the 

 fulcrum C, represented iu magnitude and direction 

 by CR, 



Let a and ft be the angles which A P and B V pro- 

 duced, make with a line CX drawn through C at right 

 angles to the vertical C Y passing through C, and tho 

 angle C R makes with C X. 



R and arc two unknown quantities which we have 

 to determine: tlir.m-h C draw CE and CD perpen- 

 dicular to A P and B W produced. 



We may regard our lever as a rigid body kept in 

 equilibrium by tho three forces P, W, and R. Resolv- 

 ing these forces into forces parallel to C X and C Y, X , 

 and Y, being the resolved parts of P, X 2 and Y,,, those 

 if \V, and X., and Y 3 those of K ; and taking inoini-nts 

 about C, we nave by Prop. XVJII., thu following con- 

 ditions of equilibrium. 



(I.) X 3 + X, X, - 0, or R cos. + W cos. ft 



p cos. o a 



