question_answer

A cable is replaced by another of the same length and material but of twice the diameter. (a) How does this affect its elongation under a given load? (b) How many times will be the maximum load it can now support without exceeding the elastic limit?

Answer:

                (a) Young's modulus,                 \[\Upsilon =\frac{Mgl}{\pi {{r}^{2}}.\Delta l}=\frac{Mgl}{\pi {{\left( \frac{D}{2} \right)}^{2}}.\Delta l}=\frac{4Mgl}{\pi {{D}^{2}}.\Delta l}\] where D is the diameter of the wire. Elongation,   \[\Delta l=\frac{4Mgl}{\pi {{D}^{2}}\Upsilon }\] i.e., \[\Delta l\propto \frac{1}{{{D}^{2}}}\] Clearly, if the diameter is doubled, the elongation will become one-fourth.  (b) Also load, \[Mg=\frac{\pi {{D}^{2}}.\Delta l.\Upsilon }{4l}\] i.e., \[Mg\propto {{D}^{2}}\] Clearly, if the diameter is doubled, the wire can support 4 times the original load.