STATICS. 



[J97- 



197. Let us try to determine the condition which 71 and T 2 must 

 satisfy to prevent slipping. To do this we determine the equilibrium of 

 the belt at the moment when slipping is just on the point of taking 

 place. 



The tension of the belt decreases gradually along the arc CGD from 

 the value 71 at C to the value T s at D. Let it be T at the point P and 

 T+ dT 'at the near point P' (Fig. 63). The portion PP' of the belt is 

 in equilibrium under the action of the forces T, T -\- dTand the reaction 



dR of the pulley ; hence dR must pass 

 through the intersection of T and 

 T-\- dT and must make with the radius 

 APan angle equal to the friction angle <. 

 Resolving these forces along T and at 

 right angles to it, we have, if %.PAP'=dO, 



T+ dR sin < = T+ dT, 



or dR sin <f> = dT, 

 dR cos < = TdO ; 



T+dT hence, dividing, 



i dT 



Fig. 63. 



Putting /u, for tan < and integrating over the whole arc of contact, we 

 find, if be the angle of this arc, 



log 71 -log T 2 = n$, 



or 



T 



For the common system of logarithms this becomes 



log * = 0.4343 /*0, 

 where must be expressed in circular measure. 



198. Rolling Friction. The resistance offered by a surface to 

 tbe rolling of another surface over it is of a somewhat different 

 nature from that of ordinary or sliding friction. In sliding fric- 

 tion, the same point or surface area of one body comes in 



