48 



ij Jli. I[(irtz:>iiliil Bir iclth sfrtfclicd suppnrtiiK/ ^prinf/ und /oai'fil iinl. 



Instead of supporting the lioriz(intal bar by a Hat s})rin^ at one end, we may 

 simply hinge it at one end to a fixed irupjiort, and hold it up by a stretched spiral 

 spring connecting another fixed siipjiort above it with some point in the 1 ar, 

 ])referablv a j»>int not far from the hinge. This arrangement has be( ii used l)y 

 Mr. Gray, wlio lias al.-<o added an ingenious plan by which tlie e.juililnium may 

 be made neutral. * 



The attachment of the supjKirting S|)ring to a ] oint near the hinge gives, of 



itself, a system poss;'ssing much more feeble stability than would be possessed by 



a mass hanging directly from a spring stretclied to the same extent. Thus, in 



the figure, let a heavy ma.<s be fixed to tlie md AliC at <", while the jioint B is 



connected with a suj)port /) by a spiral spring, and the rod is hinged at A by 



pressing against a fixed knife-edge. Let 71 be 



D ..,.,, Ar 



P^^ tlie ratio 01 lengtlis — - ■ \\ e have, m gene- 



^ ral, for a mas.s freely executing simple hanuo- 



"" fir 



R ' 



^ where M is the mass and 7^ is the whole force 



^ which tends to restore the ma.ss to its position 



^ of etiuilibrium when its di.-^placenient from 



5S tiiat ])osition is unity. To apj)ly this to the 



^ present case, we may either refer the force to 



^ the end of tiu> bar C, where the mass is placed, 



■^ ^ y- — N or rt-fer tiic mass to B, the point of attachment 



A B V_y "^ ^^^ spring. Both processes of couree lead 



to the .same re.-;ult. Taking the latter plan, 

 the mass referred to B is n'm, where m is the actual ma.ss applied at C. Tho 

 restoring force for unit displacement (of the ])oiut of reference, B) is the force 



• ■, 1 • • i- 1 ■ 1 • 1 '""'/ 



required to produce unit exteusion ot tlie spring, and is equal to — --- , wliere I 



is the actual total extension. 



nie oscillations, the ])eriod t = -~\|- 



TT 1 ' nvd nl 



Hence . r = L'r^i-- - =2--»/ — , 



which is greater in the ratio ] " : 1 tlian tlie period in which a mass would 

 oscillate if directly hung to the spring, and stretching it to the same extent. 

 The arrangement has therefore the advantage of giving more nearly neutral 

 enuilibrium than is attained by the use of a directly loaded spring of the same 

 length. On the other hand, the effective ineitia of the .system is n times less 

 than it would be if the spring were equally drawn out by a weight directly 

 applied to it. 



* Transactions of the Seisinological Society of Japan, Vol. Ill, p. 137. 



