STRING ON ROUGH CUKV! . 



the resistance on the PQ may be taken to 1> ( > a 



which acts in some direction intermediate' l..-t\\. direo- 



tions of these two normals; suppose >|r the angle which its 

 'ion makes with that of ' :ial at /'. \\'.- honhl 



<\*co8^r instead of R&s in the equations (j 

 (6), where ^ is an an.irle less than 80; hence in the limit 

 cos>|r=l and equations (4) and (8) remain nnehan 

 the term 728* sin -^r must be introduced into equations (1) and 

 (5) ; thus equation (1) becomes 



T- (T+ 8T) cos 80 - 7?8 sin ^ = ; 

 therefore -.^ 







and ultimately ^ = p and sin i|r = ; hence as before 



AT 



Similarly we may shew that equation (7) remains true after 

 the introduction of the term Il&s sim/r into equation (5). 



195. A siring is stretched over a rough plane curve; to find 

 the tension at any point and the pressure on the curve in tin 

 ng position of equilibrium. 



First suppose the weight of the string neglected. See the 

 first figure of Article 194. 



The element PQ is acted on by a tension at P along the 

 tangent at P, a tension at Q along the tangent at Q, the re- 

 sistance of the curve which will be ultimately along the nor- 

 mal at P, and the friction which will be ultimately alon# the 

 tangent at P and in the direction opposite to that in which 

 the element is about to move. Let T denote the tension at 

 /'. y+ cT that at Q, Il&s the resistance, filth the friction; 

 suppose the string about to move from A towards B. {Sup- 

 pose the element PQ to become rigid, and resolve the forces 

 acting on it along the tangent and normal at 1* ; then 



T+fjLllS8-(T+ST) cos 80 = ............ (I), 



Ilos-(T+&T) sin 80 = ............ (2). 



