189-] FRICTION. II3 



(4) A particle of weight Wis kept in equilibrium on a plane inclined 

 at an angle 6 to the horizon by a force P making an angle a with the 

 line of greatest slope (in the vertical plane at right angles to the intersec- 

 tion of the inclined plane with the horizon). Find the conditions of 

 equilibrium when the particle is on the point of moving (a) down the 

 plane, (^) up the plane. 



(5) A homogeneous straight rod AB = 2/ of weight Crests with one 

 end A on the horizontal floor, with the other end B against a vertical wall 

 whose plane is at right angles to the vertical plane of the rod. If there 

 be friction of angle < at both ends, determine the limiting position of 

 equilibrium. 



(6) Two particles whose weights are W, W are in equilibrium on an 

 inclined plane, being connected by a string directed along the line of 

 greatest slope. If the coefficients of friction are /*, /x', determine the 

 inclination of the plane. 



189. The idea of the angle of friction suggests a graphical method for 

 problems on equilibrium with friction. 



The case of a rod resting on two inclined planes, Art. 150, Fig. 41, 

 may serve as an example. If the intersection E of the normal reactions 

 A and B lies on the vertical through D, the rod will be in equilibrium 

 whether there be friction at A and B or not. When this condition is 

 not fulfilled, the rod may still be in equilibrium if there be sufficient fric- 

 tion between the ends of the rod and the supporting planes. 



Let p. = tan $ be the coefficient of friction on the plane CA, /x' = tan< f 

 that on CB ; then the total reactions at A and B will, by Art. 185, 

 make angles not greater than < and <f>', respectively, with the normals to 

 the planes. Hence the two limiting positions of equilibrium for the 

 weight W, in a given position of the rod, can be found by bringing the 

 lines of these total reactions to intersection ; the limiting position of W 

 is the vertical through this intersection. Thus, to prevent the rod from 

 sliding up the plane CA and down the plane CB., the friction angles <, 

 <' must be applied in the negative sense (clockwise) to the normals at 

 A and B ; this gives one limiting position D 1 for the point D. The 

 other position D" is found by applying the friction angles in the positive 

 sense. Equilibrium will therefore subsist if the weight be placed any- 

 where between D 1 and D". 



PART II 8 



