112 STATICS. [187. 



in other words, if the angle 9 be gradually increased, the 

 body will not slide down the plane until 6 > <. This fur- 

 nishes an experimental method of determining the angle of 

 friction $, which on this account is sometimes called the angle 

 of repose. 



187. A particle P (Fig. 57) will be in equilibrium on any 

 rough surface, if the total reaction of the surface, i.e. the result- 



/ ant R of the normal reaction 

 N and the friction F, is equal 

 and opposite to the resultant 

 / R' of all the other forces act- 

 ing on the particle. 



The limiting value of the 

 angle between N and R is </>, 

 so that the particle can be in 

 57< equilibrium only if the result- 



ant R' makes with the normal an angle <<. Hence, if about 

 the normal PN as axis, and with P as vertex, a cone be 

 described whose vertical angle is 2$, the condition of equi- 

 librium is that R' must lie within this cone. 

 The cone is called the cone of friction. 



188. Exercises. 



(1) A particle of weight W"\s in equilibrium on a rough plane inclined 

 to the horizon at an angle 6, under the action of a force P parallel to 

 the plane along its greatest slope. Determine P: (a) when 6 > <, (^) 

 when = <f>, (c) when 6 < <, < = tan" 1 /* being the angle of friction. 



(2) Determine the tractive force required to haul a train of 100 tons 

 with constant velocity up a grade of 2.5 per cent if the coefficient ol 

 friction is 1/200. 



(3) A weight W is to be hauled along a horizontal plane, the coeffi- 

 cient of friction being /* = tan </>. Determine the required tractive force 

 P if it is to act at an inclination a. to the horizon, and show that this 

 force is least when =</>. 



