20 



MECHANICS. 



proportion ; and since p is measured 

 from D, the graduation must commence 

 from that point. 



The same will be true if the counter- 

 poise be on the longer arm ; but our 

 limits compel us to leave the further in- 

 vestigation to the student. 



It is evident that there is a limit to 

 the weight which can be determined by 

 this machine. When the counterpoise 

 has been brought to that division which 

 is nearest to B, it sustains the greatest 

 weight which it is capable of deter- 

 mining. There are two ways of esti- 

 mating greater weights than this, either 

 by using a heavier counterpoise, or by 

 having another point of suspension on 

 the shorter arm nearer to the fulcrum. 

 It is evident that either will have the 

 desired effect. 



(45.) The Danish balance is a steel- 

 yard, in which the counterpoise is fixed 

 in one position, but the fulcrum is 

 moveable. It is represented in fig. 28. 



Fig. 28. 



Let C be the centre of gravity of the 

 unloaded beam, and let the distance C 

 G be x. The effort of the weight g of the 

 beam to depress the arm is then g x x. 

 The effort of W to depress the other 

 arm is W x A G, or W x (A C G C). 

 Now, let A C be called a, and the con- 

 dition of equilibrium will be 

 g x = W a W ' x 



Ol* ? 'Y* ~J~ "VV 'T* 



Wd 



(4G.) Hence maybe deduced the method 

 of graduation. The divisions must ob- 

 viously commence from C, since when 

 the loop is there it poises the unloaded 

 beam. If it be required to be graduated 

 for pounds, let the weight of the instru- 

 ment g be expressed in pounds, and let 

 a, the distance C A, be expressed in 

 inches. First, suppose W to be one 

 pound, and we have 



If we suppose W to be successively 2, 

 3, 4, &c. pounds, we have 



~ 



3 a 



For example, let a be 1 8 inches, and 

 g be 6 Ibs. the distances of the divi- 

 sions corresponding to 1,2, 3, and 4 

 Ibs. from x, will be 



18 18 36 



54 



72 



these several numbers expressing in 

 inches the distances of the several di- 

 visions from C. 



This instrument may be graduated 

 experimentally by suspending from A 

 successively 1, 2, 3, &c. pounds, and 

 -finding by trial the position of the 

 fulcrum which will produce equili- 

 brium. 



Fig. 29 



4 a c 



,T = ~ &C. 

 ff + 4 



(47.) The bent lerer balance is re- 

 presented in/g-,29. The substance to 

 be weighed is placed in a dish F, sus- 

 pended from the arm of a bent lever 

 at L. The other arm C D is terminated 

 in a heavy knob, which plays upon a 

 graduated arch G H. As the weight 

 depresses the arm C A, it is so con- 

 structed, that the leverage of W is 

 constantly diminished ; and since D is 

 moved up the arch, its leverage is con- 

 stantly increased. When D acquires 

 such a position that it counterpoises 

 the weight, the division to which the 

 index points on the graduated arch 

 expresses the amount of the weight. 



To graduate this instrument, let the 

 first division be placed at the position 

 which the index assumes when the dish 

 F is unloaded. Then let the dish be 

 successively loaded with 1, 2, 3, &c. 

 pounds or ounces, or whatever denomi- 



