An alloy wire of 2 mm² cross-sectional area and 15 N weight hangs freely under its own weight. Find the maximum length of the wire, if its extension is NOT to exceed 0.6 mm. Take modulus of elasticity (E) for the wire material as 150 GPa.

Consider an element length dx of bar at a distance x from its top end.

The weight of the bar per unit length = W/l

∴ Force on the cross-section of the elemental part \(= \frac{W}{l}\left( {l - x} \right)\)

∴ Stress on the section of the element part \(= \frac{{W\left( {l - x} \right)}}{{lA}}\)

∴ Extension of the elemental part \( = \frac{{W\left( {l - x} \right)}}{{lAE}}dx\)

∴ Total Extension \(= \mathop \smallint \limits_o^l \frac{{W\left( {l - x} \right)}}{{lAE}}dx\)

\(δ = \frac{{Wl}}{{2AE}}\)

Calculation:

Given:

δ = 0.6 mm = 0.6 × 10^-3 m, E = 150 GPa = 150 × 10⁹ N/m², A = 2 mm² = 2 × 10^-6 m², L = ?

\(δ = \frac{{Wl}}{{2AE}}\)

\(​​0.6 \times 10^{-3}= {{15 \times L} \over2 \times 2\times 10^{-6}\times150 \times 10^9}\)

L = 24 m