340 



PRACTICAL MATHEMATICS 



172. The Slipping of a Belt on a Pulley. Let a be the angle 

 of lap and T t and T 2 the tensions in the belt at P and Q respec- 

 tively (Fig. 113). 



FIG. 113. 



Considering an elementary length of belt subtending an angle 

 SO at the centre, and let the tensions on either side of this length 

 of belt be T and T + ST. 



The normal pressure N of this length of belt on the rim of the 

 pulley will be found by resolving T and T + ST in the direction OR. 



Thus N = (T + ST) cos 

 *= (2T + ST) sin 



90' 



_S0\ 

 2/ 



SO 



= (2T + ST) -?-, taking S0 as being small 



m 



= T SO, taking ST as being small 

 Force of friction = normal pressure x coefficient of friction 



= wTSO 

 When the force of friction is just equal to ST, slipping begins. 



Then 

 or 



and 

 Hence 



ST = uT SO 



M = MT 



jm 



-jjr = uT, when SO is infinitely small 



/*/"7T^ f* 



UR- = u \dQ+ Const 



log e T = wO + Const 



At P, = and T = T x log^ = Const 

 At Q, = a and T = T 2 log e T 2 = wa + Const 

 log e T 2 - log.Ti = wa 



