Question

A steel wire of length 2m and diameter 1 mm is stretched horizontally between two rigid supports spaced 2 m apart. A mass 250 g is suspended from the mid-point of the wire. Calculate the small sag x of the wire at the midpoint.

 

Young’s modulus for steel = 2×1011 N m –2. 

 

Assume that   where x<<1. The figure below might aid you in your analysis.

 

                  

Answer 1

$$T=\frac{mg}{2\sin \theta}=\frac{mg}{2x}\sqrt{0.5^2+x^2}\\T=0.25\pi \epsilon Ed^2\\x=l\left(\frac{mg}{2\pi Ed^2}\right)^{\frac{1}{3}}$$

$$x=2\left(\frac{0.25\cdot9.8}{2\pi (2\cdot10^{11})(0.001)^2}\right)^{\frac{1}{3}}=0.025\ m$$