MECHANICAL PHILOSOPHY.-STATICS. 



[VARIOUS BAT-ANCEU. 



moved and replaced by weight* until equflibruna h.e* 



been again restored. Thwe Utter weights detarmilM "i " 



that of the article ; and thus any error arising frum 



in. .p.iiity "f tin' aniii .if the balance is eliminated. 



.K "KOM itD on I;A LANCE. 



halauoe consist* of an iron or steel lever A B, 



103) with iiiu-<|ual arms A C and B C resting on a 



fulcrum with a knife-edge at C, which plays oil a pivot 



passing through a support held or bu*i>o.nded hy a ring. 



The kubntancf P whose weight is to be determined, is sus- 



.1 from tho point A by a hook, while a constant 



weight Q, called the counterpoise, is attached to a ring D, 



!i can be slipped along tin- longer ann CB. The 



edge of the ann CB U so graduated that the point D at 



h the counterpoise Q suspended from it balance* P, 



inin<-< tin- Wright i if tin- latter in pounds aud ounces, 



-.;\ other convenient iK nomination of weight for 



which the steelyard may have been graduated. 



The steelyard is sometimes furnished with a second 

 fulcrum and apparatus for its suspension, nearer to A 

 than C ; the position of the lever is then inverted, 

 ami another graduation on what was before the under 

 edge of tho arm B C, gives the weight of P for the new 

 position of tho fulcrum. The same instrument has thus 

 two different ranges of weights for the same counteqioisc, 

 the fulcrum nearer to A being used for substances lyisi-.; 

 within a range of weights greater than that for which 

 the other U graduated. 



To Graduate tlie Koman Steelyard. The Roman steel- 

 yard may be regarded as a heavy lever with unequal 

 arms resting on a fulcrum C (Fig. 104). 



Ix-t W be the weight of the lever and the hook or 

 scale-pan by which P is suspended, G the position of 

 tlu-ir joint centre of gravity, Q the weight of the counter- 

 poise, together with the ring and chain by which it is 



The steelyard " 1M - 



may now be con- Q G 



sidered as a lever 

 without 

 whose fulcrum is 

 C, acted on by 

 three parallel Q 

 and vertical for- 



n 



I 



W 



ces, P, Wand Q, at the points A, G, and D, represented 

 in magnitude and direction by the lines A P, G W, and 



A\ hen these forces produce equilibrium, taking mo- 

 ments about the point C, we have 



P A C - W G C + Q C D. 

 Hence, transposing, we have 



<)D = P AC-W GO 



orCD--^AC-~GO. 



Now let D,, D,, D,, D, &c. (Fig. 105), represent the 

 distances from C at which weights of one, two, three, 



J> 



P< 



p __ fr 



four. r any other unit of weight for which 



the steelyard is to be graduated, suspended from A are 

 respectively balanced by the counter] 



SuhMituting, t! >r 1' the nuiiilnTS 1, 2, 3, 4, 



<tc , and !>.. !>_.. l' :i , !>,,, <tc., for D in the above equa- 

 tion, we have 



W 



In A C take a point B such that B C- Q G C, and make 



AC 



\\ li.Te Q represents the weight of the 



counterpoise, in the same unit which is chos, 

 graduation, thru make D, D, B D,, D, 1), - B !>,, 

 !i II. =. B D,, ic. D,, D,, D,, D, itc, will mark 

 the points at which tho coir. ! balance 



weiu'hts of 1, 2, 3, 4, i:c., pounds sus; 111 A. 



!'. ir 



C D! = B DL - B C = A Q C - y G C. 



! BC = 



A p -ITT 



u " GC. 



C D, = B D, B C = 3 B D 4 - B C = 3 ( ;c. 



A P U 



C D = B D. B C = 4 B D! - B C = 4 u ^ GC. 



When the instrument is so constructed that the coutrc 

 of gravity G of the steelyard and scale, or hook, li 

 the vertical line passing through C, tho quantity i! C or 



W 



Q- G C becomes zero, and the points of graduation are 



taken at equal intervals from C. 



THE DANISH STKK1.VARD OR 1!ALA\CE. 

 Tlu's instrument differs from tho Roma? ,1 in 



these respects, that the counterpoise is fixed at one of 

 the extremities, while the fulcrum is movable. The 

 edge of the steelyard is graduated, and tho point at 

 which tho fulcrum is placed to cause the fixed counter- 

 ]ioise to balance the substance whose weight is to be 

 determined, marks its value. 



Fig. IOC. 



A 



CA 



TO (.': 



Dan 



-Let ]-. 

 and tho sti 

 Ijin-s P A, G '\V, 

 and DQ (Fig. IOC), 

 rep! same 



forces in this case, as in that of tho Roman steelyard ; 



in this instance, however, it is C and not D which is tho 



movable point. 

 Then taking moments of the forces as before, about 



the point C, we have 



= W(AG AC) + Q(AD AC). 

 By transposition (P + W + Q) A C = W A G 

 + Q-AD; 



, AP W-AG + Q-AD 

 P+W + Q 



Let C,, C 4 , C 3 , C 4 , c., represent the points at which 



I, 2, 3, 4, &c., pounds will lie balanced by tin: eon. 



when the fulcrum i.s placed under the 



II. nee substituting C,, C,,, C,, C t , for C, and 1, .'. . 

 <tc., for P in the preceding expression, wo have 



W-AG + Q-AD 

 AC, - 



AC 3 = 



AC, 



1 + W + G 

 W-AQ + Q-AD 



2 + W + G 

 W-AG + Q-AD 



3 + W + O 

 W-AG + Q-AD 



4 + W + G 



The reciprocals of those quantities are in arithmetical 

 progression, and therefore the distances of the point* 

 C|, C,, C s , C 4 , itc., from A will form an harmouicr.l 



'I HE BALANCE OF QUINTENX. This balanro, 

 named after its inventor Qnintens, is fre(|iien<ly em. 

 ).loyed on railways for determining; the weight ol 



and aflbrds a good example of a < ibinntion of le 



It is represented in Fig. 108. Fig. 1(>7 i.- added for tho 

 purpose of showing its mei I A 



platform A I! on which tli. i. hich we I 



weighed, is placed, is fixed in m y to ;,n upri^lit pie 



