142 Statics. 



of which the tangent is triple the radius is 71* 34'. This will 

 accordingly be the angle of friction in these cases. 



239. By means of this result, it will be easy to determine in 

 each machine what ratio ought to exist between the power and 

 the. weight in order that motion may be upon the point of taking 

 place, allowance being made for friction. 



In the lever, for example, let us suppose that the fulcrum is 

 a simple support, as represented in figure 1 28. We have seen, 

 that with respect to this machine, there cannot be an equilibri- 

 um, unless the resultant DF of the two forces;?, 7, be perpendic- 

 ular at F to the common tangent to the surface of the lever and 

 163> that of the fulcrum. On the supposition of friction, the case is 

 different ; it is still necessary that the resultant should be direct- 

 ed from the point D to the fulcrum F; but it is sufficient in order 

 to an equilibrium, that one of the two inclinations DFA, DFB, 

 according as we wish q or p to prevail, should be greater than 

 the angle of friction, and for the state of equilibrium approach- 

 ing the nearest to motion on the part of the power q, it suffices 

 that the inclination DFA should be precisely equal to the angle 

 of friction ; since, if we imagine the force according to DF de- 

 composed into two others, the one perpendicular to AB^ and the 

 other in the direction A.F, the force in the direction AF will be 

 less than the friction in the first case, and exactly equal to it in 

 the second. With respect to the two forces <?,/>, they will still 

 be in the inverse ratio of the perpendiculars FK, FL. 



240. But if the fulcrum is such that the lever can have no 

 other motion except that of a rotation, that is, if it turn on an 

 axis or pin, we adopt the following method which is common to 

 the lever, the pulley, and the wheel and axle, especially when 

 in this last machine the power and weight are in the same plane. 

 We shall consider first the wheel and axle ; the manner of treat- 

 ing the lever and pulley will afterwards appear. 



Fi 129 Let HCI be the plane of the wheel, GKL a section of the 

 cylinder, and NDM the axis about which the machine is to turn. 

 On the supposition that there is no friction, the resultant of the 

 two powers p, <?, passing through their point of meeting A, must 

 pass also through the centre F of the axis. But in the case of 



