# A steel wire of mass μ per unit length with a circular cross section has a radius of 0.1 cm.

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(a) A steel wire of mass μ per unit length with a circular cross section has a radius of 0.1 cm. The wire is of length 10 m when measured lying horizontal, and hangs from a hook on the wall. A mass of 25 kg is hung from the free end of the wire. Assuming the wire to be uniform and lateral strains << longitudinal strains, find the extension in the length of the wire. The density of steel is 7860 kg m–3 (Young’s modules Y=2×1011 Nm–2).

(b) If the yield strength of steel is 2.5×108 Nm–2, what is the maximum weight that can be hung at the lower end of the wire?

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(a) Consider an element dx at a distance x from the load (x = 0). If T (x) and T (x + dx) are tensions on the two cross sections a distance dx apart, then

T (x +dx) – T(x) = μgdx (where μ is the mass/length)

(dT/dx)dx = μgdx

T(x) = μgx +C

At x = 0, T(O) = Mg  C  Mg

T(x) = μgx + Mg

Let the length dx at x increase by dr, then

(m is the mass of the wire)