210 IN(MNKI) PLANE WITH FRICTION. 



than unity there is no other position of equilibrium; if tin- 

 coefficient of friction is greater than unity equilibrium will 



also subsist for all values of 6 between TT - '2t - nml - . 







Or the results for the third case may be deduced from those 

 i in tin second case, observing that <x = e. 



JSquiJtLritim of Machines with Friction. 



178. Inclined Plane. 



Let a be the inclination of the plane to the horizon. Sup- 

 pose a force P l acting at an in- 

 clination 6 to the plane and the 

 body on the point ot moving down 

 the plane. Let R be the normal 

 action of the plane, pR the friction 

 which acts up the piano, W the 

 weight of the body. Resolve the 

 forces along and perpendicular to 

 the plane; then, for equilibrium 

 we have 



=0 (1), 



= (2). 



Substitute in (1) the value of R from (2) ; thus 



_ J r sin a ft TT cos a 



1 ~ 6 -p sin 6 * 



Next, suppose P s a force acting at an inclination 6 to the 

 plane, such that the body is on the point of moving //// the 

 plane. Friction now acts down the plane, and we shall iind 



p W &m a + pW COB a 



2 ~~ cos 6 + fj, sin 6 



This result may be deduced from the former by changing the 

 sign of p. 



re will be equilibrium if the body be acted on by a 

 force P, the magnitude of which lies between P l and P s . 



