We write the equilibrium condition for a small segment of the string which had the length
Δx before suspension and was at a distance x from the point of suspension (Fig. 149):
where L is the length of the rubber string in the unstretched state. Thus, it is clear that after the suspension the tension will uniformly decrease along the string from mg to zero.
Therefore, the elongations per unit length for small equal segments of the string in the unstressed state after the suspension will also linearly decrease from the maximum value to zero. For this reason, the half-sum of the elongations of two segments of the string symmetric about its middle will be equal to the elongation of the central segment which experiences the tension mg/2. Consequently, the elongation Δl of the string will be such as if it were acted upon by the force mg/2 at the point of suspension and at the lower end, and the string were weightless; hence
