74 THE HUMAN MOTOR 



applied to the load imposed on the plane, it can only be pulled 

 by a force superior to F + 3>. The useful component of the 

 tractive force is P cos p, parallel to the plane. The pressure 

 (perpendicular) N is diminished by the perpendicular component 

 of P, which is P sin p. As N = Q cos a, the pressure of the load 

 will be Q cos a P sin p, and the friction 3> = / (Q cos a P 

 sin P). 



Finally the tangential component of the resistance is Q sin 

 a = F. Therefore 



p cos p = Q sin a +/ (Q cos a P sin p). 



From this 



sin a +/ cos a ^ sin (g+y) 



" ~ cos p +/sin p ^ cos (p 9) 



P is a minimum for the maximum value of ccs (p 9), or 

 P = 9. Therefore P = Q sin (a + 9) ; it is necessary for the 

 pull to be applied in a direction making the angle (3=9 with the 

 inclined plane (fig. 90). 



To allow the load simply to slide on the plane by the action 

 of gravity, consider the equilibrium between the component 

 F == O sin a and the friction $ =/N =/Q ccs a. The force of 

 sliding friction will be : 



F & = Q (sin a / cos a) . 



Replacing /by - , it would be written 

 J cos 9 



cos 9 



According to whether a be smaller or greater than 9, the move- 

 ment would be retarded or accelerated. 



The inclined plane is used in small quarry railways, in stations 

 and shops to move packages, and for drays to be loaded with 

 barrels, etc. 



The useful power being P cos p and the resistance Q sin a for a 

 virtual displacement of the body, /, the equality of the work 

 depending on motion and resistance respectively is expressed : 

 PI cos p = Q/ sin a. 



If / is the total distance covered AB, then 



AC = AB sin a = / sin a = h ; 

 so that : 



Q/ sin a = Qh, 



( l ) / has been replaced by and simple transformation effected (see 



cos <p 

 any text-book of trigonometry). 



