On the Motion of Bodies affected lij Friction. 1 IS 

 center at the beginning of the motion, we shall have 

 the A\ hole rotatory motion ; hence, therefore, t- + L? x 



a -^ '^ a- — 2 F « iii ^ a- — 2 F x, consequently z i 

 the space described before the 



motion becomes uniform. 



2. If this value of z be substituted into the expression 



for the velccitv, we shall have ^— ^ for the ve- 



as 



locity when the friction ceases. 



3. If f = fl, then x = o, and hence the body will con- 

 tinue to move uniformly with the first velocity. 



4. If V be greater than a, then the rotatory inotion of 

 the point a on the plane being greater than its progressive 

 motion and in a contrary direction, the absolute motion of 

 the point a upon the plane will be in the direction ML, and 

 consequently friction will now act in the direction LM in 

 which the body moves, and therefore will accelerate the 

 progressive and retard the rotatory motion ; hence it appears 

 that the progressive motion of a Indy may be Accelerated 

 ly friction. Now to determine the space described before 

 the motion becomes uniform, we may observe that as the 

 progressive motion of the body is now accelerated, the ve- 

 locity after it has described any space z will be = V a + 2 Yzl 

 hence the velocity acquired ^ ^ a- -\- ^¥ z — a, and con- 



sequently the rotatory velocity destroyed — X Va-' + 2F3; — £7, 



hence v • x V' a- + 2 F x — a = ^ a" + 2 F x, there- 



_ r5 X i' + z" " X a — fl- x fli- , . , 



lore z — — • the space required. 



2 1* X ai'^ '^ ^ 



3. If a = 0. or the body be placed upon the plane with- 

 out Anv proorcssive velocity, then 2; = — . 



' ^ => •'' 2Fxai"- 



Cask III.— 1. Let the given rotatory motion be in the 

 direction gef-, then as the friction must in this case always 

 act in the direction ML, it nmst continually tend to destroy 

 bolh the progressive and rotatory motion. Now as the ve- 

 locity destroyed in the same time is in proportion to the re- 

 tarding I'orce, and the force which retards the rotatory is to 

 the force which retards the progressive velocity by Cor. 2. 

 Prop. r. us ru : r s, therefore if v be to a asi ra 1% to r v, 

 H 'i then 



