BALANl ES. ] 



MECHANICAL PHILOSOPHY. STATICS. 



721 



(2). Y n Y 3 T, = 0, or R sin. 6 W sin. ft 

 1' sin.'n = 0. 



And(3). P-CE W-CD=0. 

 From this. last equation we obtain P C E = W' C D, or 

 P CD Perpendicular on direction of W. 

 W C E Perpendicular ou direction of P. 



From equations (1) and (2) we can determine R and 9, 

 and consequently the magnitude and direction of the 

 pressure on the fulcrum, which will be equal to R, and 

 act in the direction opposite to C R. 



Transposing equation 



(1). R cos. = P cos. a W cos. ft 

 (2). R sin. = P sin. a + W sin. ft 



Dividing one of these equations by the other, we have 



n P sin. a + W sin. ft 

 tan. v = T. ... > 



P cos. a W cos. ft 



which determines 0. 



Or, adding their squares, we have 



R 2 (cos. 2 + sin. 2 9) = P 2 (cos. 2 a + sin. 3 a) + \V- 

 (cos. 2 ft + sin. 2 ft)- 2 P W (cos. a cos. fl-sin. a sin. ft). 



Hence, by Trigonometry, 



R 2 = P 2 + W 2 - 2 P W cos. (a +ft), 

 which determines the magnitude of R. 



If we had supposed B instead of C as the point of 

 lii-ation of the fulcrum, and token our moments 

 . t. B, we should have Lad 



P Perpendicular on direction of R 

 11 "Perpendicular on direction of P 



Hence Uie condition of equilibrium in a lever of any 

 kind, that the power mutt be to the weight, inversely as 

 t/'f /i?rj>endicular$ drawn from the falorum on their direc- 

 tion!. 



The bent lever evidently includes the straight one, 

 as in the latter case A, B, and C are in the same straight 

 line'. 



THE I'.KNT I.KVKR BALANCE. The bent-lever 

 balance (Fig 98) is a machine which, within certain 

 limits, enables u.t to weigh substances without the use 

 of weights ; it consists of a rig. 98. 



bent lever whose two arms A 



are A C and U C, movable 

 about a fulcrum C. The ful- 

 crum C is fixed to a stand 

 which carries a graduated 

 arc ; over this arc the extre- 

 mity B of the lever moves 

 as C B turns round the ful- 

 crum C. From the other 

 extremity A, a scale-pan E 

 is go suspended as to have 

 its centre in all positions of the lever in the vertical 

 passing through A. 



A weight D, is fixed to the arm CD so as to bring 

 the centre of gravity of the whole lever and scale-pan to 

 Fig. 99. 



mme point below thn fulcrum C : the magnitude of thii 

 vol. i. 



weight is so arranged that the extremity B of the 

 lever shall point to zero on the graduated arc, when 

 the scale-pan is empty, and the lever is in a state of 

 equilibrium. 



To graduate the arc, weights of 1, 2, 3, &c., pounds 

 or ounces, or whatever denomination of weight the in- 

 strument is intended to indicate, are placed successively 

 in the scale-pan ; and the corresponding points of the 

 arc over which B rests, are marked on the scale as 1, 2, 

 3, ,Vc.- 



This balance is of great use for determining quickly 

 the weights of bodies where extreme nicety is not 

 essential. 



To explain the graduation of this balance mathema- 

 tically, 



Let C (Fig. 99) be the fulcrum of the lever, A C and 

 B C its arms ; let G in B C be the centre of gravity of 

 the whole lever, exclusive of the scale-pan and wire by 

 which it is suspended. Let W represent this weight, 

 and let C A = o and C G = 6. Also let S represent the 

 weight of the scale-pan and wire by which it is sus- 

 pended, P that of a body placed in the scale-pan. 



Let 9 be the angle CB makes with a line DCE 

 drawn through C perpendicular to CF, the vertical 

 passing through C ; ft the angle A C makes with 

 DCE. 



Also let a = B C A be the anele the arms of the lever 

 make with each other, and let <t=180 , and let D and 

 E 1 t" points where the vertical lines passing through 

 Aa-'v -ut the line D CE. 



Then Prop. XXIV. (P+S) C D=W C E, 

 or (P-f S) a cos. ft= \V 6 cos. 9. 

 but ft = 90 - 0+a = 90- (0 - a); hence cos. = 

 sin. (0-a). 



Therefore (P+S)a sin. (0-<i)=W& cos. 0, 



or (P-f-S)a {sin. 0cos. a cos. 8 sin a} = \V6 cos. 0, 



and tan. cos. a sin. a =, , ,, ' 

 (F-f-b)a 



or tan. = 75-7 ,+ tan. a. 



(P-r-S)a cos. a ' 



Hence if we wish to graduate the arc for pounds, 

 we must take P successively equal to 0, 1, 2, 3, <fec., 

 pounds, in which case tan. will be successively equal to 



W-6 W6 



= - ;+ tan a, . . ,. ,-f-tan. a. 

 Sa. cos. o ' ' (S-fl)a. cos. o ' 



W6 , W6 



n a, ,.. i - -,+tan a, <co., 

 ' (S+3)a. cos. a ' 



cos- a 



where W and S represent the weight of the lever and 

 scale-pan in pounds. 



THE COMMON BALANCE. This instrument is 

 popularly called the scales, or a pair of scales, and is per- 

 liaps more frequently used than any other for determin- 

 ing the weights of substances or goods. It consists of a 

 lever supported on an axis or fulcrum equally distant 



Fig. 100. 



'rom its two extremities ; under each of these extremities 



4 z 



