STRINGS 77 



55. In the more general use in which the contact is not perfectly 

 smooth, let fi be the coefficient of friction, so that p = tan e, then 

 equation (13) may be written 



dT 



and integrating this, = d 



d (log T) = d (fJL0), 



or log T = pd + a constant. 



Let T be the tension at A, then we find, by putting 6 = 0, that 

 the constant must be equal to log T ot so that 



or T= T e . (14) 



If the string leaves the surface again at some point B at which 

 the normal makes an angle ^r with the normal at A, we find for 

 the tension at B 



so that the tension is multiplied by e^ on passing over the surface 

 from A to B. 



If the string (or rope) is passed round and round a post or bollard, 

 the tension is increased in the ratio e 2 ^ for each complete turn. 

 For a hemp rope on oak the coefficient of friction, according to 

 Morin,is/* = 0.53. Thus 2/A-Tr = 3.34 and e 2l " r = 28.1. The tension 

 of a rope wound round an oak post is accordingly increased about 

 twenty-eight fold for each complete turn. 



EXAMPLES 



1. A weight is suspended by a rope which, after being wound round a hori- 

 zontal beam, leaves the beam horizontally, its end being controlled by a work- 

 man. If the rope makes 1| complete revolutions round the beam, what force 

 must be exerted by the man 



(a) to keep the weight from slipping ? 



(b) to raise the weight ? 

 (Take /* = .) 



