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.
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