44 FOECES ACTING ON A SINGLE PAETICLE 



It will be proved later that when a perfectly flexible, weightless 

 string passes over a smooth peg or pulley, the tension has the same 

 magnitude at all points of the string, and at points of contact with the 

 peg or pulley its direction is along the tangent to the peg or pulley. 



37. If the string is not absolutely weightless, but is very light, any 

 particle such as q will be acted on by three forces, its weight vertically 

 down, and the two forces from the adjacent particles acting along pq, 



rq. By Lami 's theorem , each force 

 must be proportional to the sine 

 of the angle between the remain- 

 ing two forces. Since the weight 

 . 19 is small, sin pqr must be small ; i.e. 



pqr must be very nearly a straight 



line. The line cannot be perfectly straight, however, unless the string is 

 absolutely weightless ; thus in a real string there must always be a certain 

 " sag," due to the weight of the string, although this sag may be so slight 

 as to be imperceptible. 



38. Extensible and inextensible strings. The tension, as will 

 have been seen, is a force acting at every point of the string, and 

 tending to stretch the string in the direction of its length. The 

 string either may or may not yield to this tendency to stretch. A 

 string which stretches under tension is called extensible ; a string 

 which does not stretch at all, or which stretches so little that the 

 amount of stretching is inappreciable, is called inextensible. 



Thus an inextensible string remains of the same length what- 

 ever tension is applied to it, while the length of an extensible string 

 depends on its tension. 



In 1660 Hooke discovered a law which expressed a relation 

 between the tension and the amount of stretching in a string : the 

 one is proportional to the other. 



DEFINITION. The length of a string when the tension is zero is 

 called the " natural length " of the string. 



DEFINITION. The amount by which the length of a stretched 

 string exceeds the natural length of the same string is called the 

 "extension" of the string. 



HOOKE'S LAW. The tension of a string is proportional to the 

 extension. 



