32 



and, if there were nu frictimi, tlie reeords given l)y a tracer fixed to the block 

 •■ihould he reduced by this amount. In Dr. A'^erljetk'.s instrument the mass of 

 tile block was .so great relatively to the mass of the balls that the impulse which 

 their inertia gave to the block- niu.*t have been negligilile: in other words the 

 steady jlane must have been exceedingly elo.se to the top of the balls. The 

 effective inertia of the system (with three balls) is 



3 (m + l^ M r 

 l^ + lM 



§ 32. Friciioii ill the Holliii;/ Sphere Seifinometer. 



Frictiiin, or rather rolling resistance, constitutes a serious source of error in 

 the indication.^ and has of course the ellect of making the I'ecorded displacements 

 smaller than they should be by an amount dejiendiug, in a given instrument, on 

 the nite of acceleration of the earth's surface during the disturbance. According 

 to Rankine (A|iplied Mt'chanics, ^ ()S2) the rolling resistance may betaken as a 

 c(iii|ile who.se moment is fnuiiil liy niulti])lying the normal pressure between the 

 rclling surfaces by an arm whose length (which we will write 1} depends on the 

 nature of the surfaces. To simplify matters we may consider .separately these 

 two cases: (!) mass and weight of lialls negligible compared with block, a.s in 

 \'erb(ck'.^ a](];aratus; (2) no bloek, tlie inertia and jirc-^sure being due to a .spheri- 

 cal ball alone. 



In ( 1 ), calling tin- nuiss of tlie wliole block M, the ]pressure at the top and 

 alr-o at the bottom i^if the lialls is Mi/ ancl the couple due to rolling resistance at 

 the.<c two places is {I -j- /') M./, when' / and T apply to the two pairs of surfaces. 



This is eipiivah nt to a horizontal force -— '—- a])plicd to the block, and 



^ I' 



, . I' -!- ''>'/ 

 produces an acceleiation - - -' • 



_ )■ 



In ( 2 I, calling tn the mas.« of the ball, the elective inertia (or mass referred 



to the .-teady pointl is iiii, and the couple due to rolling resistance is Imrj. 



... I Hill ... , ■ . , , 



iliis IS c(|Uivalent to a force - . - ajipiied at tlie steady pomt, and produces a 



horizontal acceleration there eipial to ' • Hence it appears that Ixith arrange- 



ii'.ents are atfeeled by rolling friction (o (he same extent, ]>i'ovided we use balls 

 of the .''^ame radius in both, and the same twn materials at each place where 

 rolling contact occnrs. 



Iiankine gives the following \alues for the aim / expies.-^ed in feet: — 



Oak on oak O.tKMi (Coulomb) 



Tjignum-Vitai on oak O.OOf ,, 



Ca.-^t-iroii on east iron (».()()•_' (Tredgold). 



If we >iip]iosc that bv ehiiosiiig .-uitable substaneis for fixed slab, balls, and 

 block, the value of / and I' can lie reduced to, .say, half of the least of these, or 



