123-125] EXAMPLE OF STRESS. 119 



125. Tension of a string or chain. In Mechanics a string 

 is regarded as a mere line of particles, but the mass of any 

 segment of the string of finite length is supposed to be finite. 

 When the mass is proportional to the length the string is said to 

 be uniform, but when this is not the case the ratio of the number 

 of units of mass in any segment to the number of units of length 

 in the segment has a finite limit when the length of the segment 

 is indefinitely diminished, and this ratio is called the line-density 

 or mass per unit length of the string. 



A plane drawn across a string, at right angles to the tangent 

 to the line of the string at the point where the plane meets it, 

 is called a normal plane. The resultant stress across the normal 

 plane is conceived to be a single force at the point where the line 

 meets the plane, and directed along the tangent. This stress is a 

 tension, (Article 122) and is called the tension of the string at 

 the point. 



Any portion of a string between two particles P, P moves 

 under the action of the external forces on its particles and the 

 tensions at its ends. The motion of its centre of inertia depends 

 on these forces only. 



In particular if P is very close to P, and if S is the sum of the 

 resolved parts parallel to the tangent at P (in the sense PP ) of the 

 external forces applied to the particles in the segment PP , p the 

 line density, As the length of PP , f the acceleration of the 

 centre of inertia of PP in the direction of the tangent at P, T 

 the tension at P, T the tension at P , and A$ the angle between 

 the tangents at P and P , we have the equation 



Now when P approaches indefinitely close to P the expression 



T cosAci-r ... , .. ., ,. , . dT , , . ,. . ... , 



. *- - will have a limit which is ,- (cf. Article do), and 

 As as 



a 



will have a finite limit which is the acceleration produced in 



a particle at P by the external forces. Suppose that the external 

 forces are bodily forces, and suppose that this acceleration is F. 

 Then our equation can be written 



