40 



§ 38. Influence of friction on the Fendulnni ami other Sei.-onomderg. 



Before going further, it may he useful to enquire more particularly into the 

 effect of frictional resistance on a i)endulum seismometei-, in order that we may 

 estimate the amount of inaccuracy whidi is intnxlucod when enough friction is 

 added to niake static records practicable, hy preventing the j)pnduluni from 

 swinging. The theory is in fact applicable not only to the common pendulum, 

 but to all seismometers whose stability would (in the absence of friction) eaus<' 

 them to execute simjile harmonic oscillations. 



Let Mhe the effective inertia of the in.strument — in other words the mass 

 referred to the steady point. Let Fhe the force per unit of that mas.« (due to 

 stability, and leaving friction out of account | which tends to restore the mass to its 

 position of equilibrium whrn tlie steady point is displaced througli unit distanci' 

 from tiiat position. Then, by the assumed condition, the total restoring force when 

 the displacement is >• is equal to r il/ F. Also, let / be the frictional resistance 

 referred to the same jjoint. This, which is made up of forces at the point or axis 

 of suspension, at the fulcrum of the multiplying pointer, at the joint by which the 

 pointer is connected to the main m;x,ss, and at the marking pointer, will, in gene- 

 ral, be a constant force, sensibly independent of the velocity. I^et the mass be dis- 

 placed so that the di.splacement of the steady point from its position of «juiiibrium 

 is r„. It will swing back, and over to tlie other side to a distance of, s;\y, r^ from 

 the jwsition of e<juilibrium. During tliis motion the energy dissipated tlirough 

 friction is/tr,, + r,). This must be equal to the loss of jwtential energy wliich is 



X 



rMh dr, or 



o 

 2/ 



Hence ,_,_=^^, 



and the same expression will give the decrease in amjilitude of displacement 



during every succeeding half-swing. The amplitudes diminish in arithmetical 



4 f 

 progression, successive displacements to the same side differing by .,"„ • 



From the above equutittn 



{r,-,; ) FM 

 2 



we have — - — —- for tiie acceleration whidi the frictional resistance pro- 

 duces whenever it acts. F, if not dii-ectly calculable, is easily determined by 

 observing the period of free oscillation of the suspended mass,* and ?■„ — r, 

 may be at once determined by cxiieriment in any actual instrument. 



An earthquake in whicli tlic acceleration of tlie ground did not exceed this 



( ,. r)F 



quantity, — ^^ .)^' — ' would ocrur without affecting the instrument, and would 



2 n- 

 Since th(> complete iierioil of fret' oscillation is — =, 



