i U On the Motion of Bodies affectedly Frictiort. 



carried forwards by the latter ; consequently the point of 

 contact of the body with the plane will then have no mo- 

 tion in the direction of the plane, and hence the frictioit 

 will at that instant cease, and the body will continue to roll 

 on uniformly without sliding with the velocity which it has 

 at that point. Put therefare z z^ the space described from 

 the comniencen^nt of the nrotion till it becomes uniform, 

 then- the body being uniformly retarded, th<; spaces from 

 the. end of the motion vary as the squares of tlie velocities, 



hence ~ : a" t; t : qF). :: —^ — x. : a- *- 2 F 2 =s 



square of the progressive velockv when the motion r>ecome3 

 uniform ; therefore the velocity destroyed by friction = a — 

 ■^ a- — 2 F« ; hence, as the velocity generated or destroyed 

 in tlie same time is in proportion to the force, we have 



l>y Cor.- 2. Prop, l.rs ; ra :: a— s/a'-—sFz : — x 

 cj -_ \/c^2 ^2 F z the velocity of the cwcKonference ejg ge- 



nerated about the center, consequently \^a- — 2Fz= — x 



c -^ V a-: _ 2 Fa;, and hence z = ^, tne 



a s- X 2 1' 



Space which the body cIcsGribcs before the motion becomes 



uniform , 



2. If we substitute this value of z into the expression for 



the velocity, we shall have ex- for the velocity of the 



" ' rs ' 



bt)dv when its motion Ix'comes uniform ; hence therefore it 

 appears that the velocity of the body, when the friction 

 ceases, witt be the san>e whatever be the tpantity of the 

 friction. If the body be the circmnferersce of a circle, it 

 will always lose half the velocity l>efbre its motion becomes 

 uniform. 



(;\seII.-— 1. Let the badv, besides having a progressive 

 relocity in the directimi LlSt (tig. 3.) have also a rotatoiy 

 motion about its center in the direction gfe, and let v re- 

 present the initial velocity of any point of the circumference 

 about the center, and suppose it first to be less than a ; then 

 friction being a uniformly retarding force, no alteration of 

 the vcloeitv of the point of contact of the body upon tl\e 

 plane can aflcct the quantity of friction ; hence the pro- 

 gressive velocity of the body will be the same as before, and 

 consequently the rotatory velocity generated by friction will 

 iil»o Ik- the samej to which if we add the velocity about the 



center 



