VARIOUS BALANCES.] 



MECHANICAL PHILOSOPHY. STATICS. 



725 





framework B C. B C again is attached rigidly to an ob- 



Fig. 107. 



lique piece DC fixed in D, as shown in Fig. IC'T, so that 

 the whole portion A B C D forms one rigid and inflexible 

 body. This rigid piece A B C D is supported at E by a 

 kuife-edge fulcrum, resting on the bar or lever F G, and 

 at D by a rod H K, which is firmly fixed to D, and sus- 

 pended by a hook at K from the lever L M N. 



Fig. 108. 



N 



Let the weights P and Q be represented in ma" aituda 

 and direction by N P and S Q (Fig. 109.) 



The weight of the substance placed on the platform 

 A B being equal to a single force S Q passing through 

 the vertical which passes through its centre of gravity, 

 will be balanced by the reaction of the pressure B,' 

 which it produces on the fulcrum E, represented in mag- 

 nitude and direction by ER,, and the reaction R 2 of 

 the tension it produces on the rod H K. 



The pressure R, of the fulcrum E on the lever F G is 

 kept in equilibrium by the reaction of the pressure it 

 exerts on the fulcrum F, and the reaction R a of the 

 tension it produces on the rod G L. 



Finally, the lever L M N, resting on the fulcrum M, 

 in acted on by three vertical forces P, R 2 and R 3 , acting 

 on the points N, K and L. 



For the equilibrium of the rigid body A B C H we 

 have as a condition 



OrR, 



For the lever F E G taking momenta about E we have 

 R,-EF = R 3 -FG. 

 FG 



''EF' 



Lastly, for the lever L M N taking moments 

 about M we have 



P M 



= R., M K + B, L M. 

 , MN " , _ LM 

 Or P -.-,- =R 2 +R 3 -, r . r - 



T> * LM 



Batgg 



machine. 



MK 

 FG 



= EF b ' 



MK 

 construction of tli3 



Hence P ' M ,r =R 2 + R 3 ' 



EF 



R 1 =Q. 



The whole weight, therefore, of Q and the framework 

 A I: i ' I) rests on the points E and K. 



The lever F G is supported by a fulcrum at one ex- 

 tremity F, while the other extremity G is suspended by 

 a rod L G, hanging at L from the extremity of the lever 

 LMN. 



Lastly, the lever L M X is supported by a fulcrum M, 

 which is fixed to the framework of the machine ; tlio ex- 

 tremity N of this lever has a scale-pan suspended from 

 it, in which weights may be placed. 



The points E, L, and K are so chosen that - = 



To avoid the consideration of the weights of the 

 various parts of this machine, we will suppose that the 

 weight of the scale-pan and the length of the arm M X 

 of the lever LMX have been so chosen, as to produce 

 an equilibrium of all the parts of the machine, and to 

 keep L M N in a perfectly horizontal position, when no 

 weight is placed on the platform A B. 



To determine the weight P, which must be placed in 

 the scale-pan suspended from N to balance a substance 

 of a given weight placed on the platform A B, we will 



Fig. 109. 



P MK 

 And = 





Rj 



_, 



! 

 j 

 YRi 



'nc the conditions of equilibrium for the several i 

 parts of the machine. 



If M N be taken ten times as long as M K, we have 

 M X = 10 M K and Q = 10 P ; and in this caso a woi.^lit 

 the tenth part of Q, when placed in the scale-pan, will 

 produce equilibrium. 



ul of placing weights in the scale-pan P (Fig. 

 108), the arm of the lever M N may be graduated, and 

 used as a Roman steelyard. 



Before placing a body on the platform A B (Fig. 108) 

 whose weight is to be determined, it is necessary to 

 observe whether the arm of the lever M N is in a pei- 

 fectly horizontal position. This is indicated by a hori- 

 zontal pin 6 fixed to the framework of the instrument, 

 being in the same line with a similar pin fixed to the 

 ann of the lever. To bring these points into this position, 

 small weights are added to or taken from a little cup a, 

 placed under the point from which the scale-pan is 

 suspended. 



ROBERVAL'S BALAXCE. Many of the balances 



used in shops are constructed on the principle of this 



machine, which is interesting for its paradoxical 



character. It is apparently a lever, on the arms of 



-i which if two weights balance each other, they will still 



continue to do so from whatever points in those 



arms they may be suspended. The accompanying 



p diagram will give an idea of the. construction of tin's 



machine. 



It consists of four bars, A B, CD, AC, and 

 BD(Fig. 110); AB being equal to CD, and AC 

 to B D. These four bars are united together by 

 four pivots, A, li, C, and D ; they also rest upon 

 two pivots or axes E and F fixed to a,n upright bar 

 E F, resting on a firm base G H. 



The holes for the pivots E and F are so placed in 

 A B and C D that B K is equal to D F. 



Lastly, two bars KL ami MX arc fixud firmly 

 at right angles to A C and B D, so that K L and A (J 

 form one rigid piece and M N and B D another. 



