On the Motion of Bodies affected hj Friction. 57 



to the poiiit of suspension a * ; then, because no force 

 actn:<j; at a can aflect the motion of the point s, that point, 

 nolw'fthstanding the action of the friction at a, will always 

 have a motion parallel to CA uniformly accelerated by a 

 force equal to that with which the body would be accelerated 

 if it had no friction; hence, if 2m= 32^ feet, the velo- 

 city acquired by the point s in the first second will be = 



llf2l£^: now the excess of the velocity of the point s above 



that of r (r being the ccnt(n-) is manifestly the velocity with 

 which s is carried about r ; hence the velocity of s about 



the center =:if^-2« = if2li^*.^^^i^, consequently 



2'?.'XCB— iflxCA ZT?,' XJ-rt xCB — i^XrrtX CA 



rs :ra:: -^ : I-T^Tca: ~ 



= the velocity with which a point of the circumference is 

 carried about the center, and which therefore expresses the 

 force which accelerates the rotation ; now as 2 a expresses 

 the accclerative force of the body down the plane, and the 

 spaces described in the same time are in proportion to those 

 _ . ^ „. 2 wx ''^X^-B — ^'^Xf « X (-'A. 

 lorces, we have 2 a : CA :: -— ■ : 



^;;Xr/^XCB-flX>-^X CA ^^^ ^p^^^ ^.j^j^j^ ^^^, ^^-^^^ ^f the 



''X''-s 



circumference describes about the center in the whole time 

 of the body's descent down CA ; which being divided by 

 the circumference p xr a (where p = 6.283, &c.) will 



(rive "''X-'^^ — "^^'^ for the whole number of revolutions 



required. 



Cor. 1. If a X CA = w X BC, the number of revolu- 

 tions = 0, and therefore the body w ill then only slide ; 

 consequentlv the friction vanishes. 



Cw. 2. Let a'// (fig. 2.) be the next position of ar s, 

 anidraw ///; parallel to s a, then will /t represent the 

 retardation of the center r arising from friction, and a' ^ 

 will represent the acceleration of a point of the circumfe- 

 rence about its center ; hence the retardation of the center : 

 acceleration of the circumference about the center :: s' t : 

 a I' :: (by sim. a's) t r : h/ :: rs : r a. 



Cor. 3. If a' coincides with a, the body does not slide, 

 but only roZ/; now in this case 5/ : r r :: as : ur; but 



• rt and 5 are not fixed points in the body, but the former alway^i rc- 

 presmts that point of the body in contact witli the plane, and the latter 

 the corrtiF'^fiding centet of oscilUiion. 



US 



