A mild steel wire of length 2 L and cross-sectional area A is stretched, well with in the elastic limit, horizontally between two pillars as shown in figure. A mass m is suspended from the mid-point of the wire strain in the wire is

Step-by-step solution

Increase in length Δ   L = B O + O C - B C = 2 B O - 2 B D ⇒   Δ L = 2 B O - 2   L = 2   L 2 + x 2 1 2 - 2   L or Δ L = 2   L   1 + x 2 L 2 1 2 - 2   L [ ∵ using Bionomial theorem] ⇒   Δ L ≈ 2   L   1 + 1 2 x 2 L 2 - 1 = x 2 L ∵   x < < L ∴ Strain = Δ L 2   L = 1 2   L × ∆ L = 1 2   L × x 2 L = x 2 2   L 2 .