18 



S lil. Ikfeniiliialion nj tlic Eß'tctirc Iiifilin and Sttadi/ Line. 



Ill est i uniting the cifective inertia and stcatlinets of a system composed of a 

 frame, or piece jiivottcd to tlie earth, and a l>>b, or piece ])ivottcd to tlie frame at 

 its corresjiondino; steady axis (|iassini>; throngii tiie centre of ])erciission), it is 

 coiiyeiiient to remember tluit tlie fiaiiie i-< ciiiiiyalent, kiuetically as well as 

 statically, to a jiair of masses, one coneentrate<l at the axis of support and the 

 other at the centre of percussion. 

 Calling 



My tlie mass of the frame ; 

 Mj tlie mas« of the hoi) ; 



r the distance of the centre of grayity of M^from the axis of support; 

 r' the distance of the centre of percussion of the frame from the axis 

 of su j)port; 



7' 



thiii iNI, is e(juiyalent to a mass — ^ M at the centre of percussion, together \yith 



ss |1 ,) Mf at the axis of support. Of these the former alone coutri- 



a mass 



hutes to the steadiness of tlie system. The hob is equiyalent to a particle of 

 mass M,, concentrated at the axis about which it is free to turn on the frame; and 

 till' «liiilc clVfctiye inertia (or in other words the mass referred to the steady line) is 



^h + \ M,. 



r 



If iii>-t<a<l of iiciiig jiiyotted the bob were tixed t" the IVaiiie, we should require 

 totnat the whole mass as "frame". The effect wmild be to shift the steady 

 line outwards, since the moment of inertia of the system about the axis of su])port 

 would be increased. The difl'erence in general effect between a frame ^vith 

 ])iyottc<l bob and a frame consisting of sui)])orting ])iece and boh rigidly connected 

 is consideiiilile if the transyerse dimensions of the bob are comparable to its 

 distance from the axis of support, and in that case we do well to piyot the bob; 

 but when the bob is small and the horizontal length of the frame is great, 

 piyottiiig the bob biiiigs littli^ advantage to counterbalance the com])lication of 

 jiarts which it inyolyes. 



A conyeiiient exjierimental way of Unding the steady line of the frame is to 

 hang it up yerticaily, with the ordinary axis of supjiort ]ilaced horizontally. Then 

 let it be swung through small arcs as a coinniou ]ieiiduliim, ami its ]ieriod (r) 

 measured, 'i'lie eiiuiyaleiit simple ))cn(luluni is one whose length is v' , the 

 distance from the axis of support to the centre of jienussion, and therefore 



•■' = -&• 



§ 20. MuHiplicaticn and .L'<'i/i.'<tri/ af the mution. 



In those eaitlii|iiakes which both fioiii their fi'e(|reiicy and moderate charac- 

 ter are best suited tor rci;istration, the actual motion.- arc u^uallv so small that to 



