A rubber rope of length 8 m is hung from the ceiling of a room. What is the increase in length of rope due to its own weight? (Young's modulus of elasticity of rubber = $5\times {10}^6 {\mathrm{Nm}}^{-2} \mathrm{and} \mathrm{density} \mathrm{of} \mathrm{rubber} = 1.5 \times {10}^3 \mathrm{kg} {\mathrm{m}}^{-3}. \mathrm{Take} \mathrm{g} = 10 {\mathrm{ms}}^{-2}$)

Y = $\frac{\mathrm{Mg}}{\mathrm{A}}\times \frac{{\displaystyle \frac{\mathrm{L}}{2}}}{∆\mathrm{L}}$

( Length is taken as $\frac{\mathrm{L}}{2} \mathrm{because} \mathrm{weight} \mathrm{acts} \mathrm{at} \mathrm{C}.\mathrm{G}.)$

Now, $\mathrm{M} = \mathrm{AL\rho }$

[For the purpose of calculating mass, the whole of geometrical length L is to be considered]

$\mathrm{Y} = \frac{\mathrm{AL\rho gL}}{2\mathrm{A}∆\mathrm{L}}$

or $∆\mathrm{L} = \frac{{\mathrm{\rho gL}}^2}{2\mathrm{Y}} = \frac{1.5\times {10}^3\times 10\times 8\times 8}{2\times 5\times {10}^6}\mathrm{m}$

         = 9.6 $\times {10}^{-2} \mathrm{m} = 9.6\times {10}^{-2}\times {10}^3 \mathrm{mm}$

          = 96 mm