  Question

If the work done in stretching a wire by $$1mm$$ is $$2J$$, the work necessary for stretching another wire of same material but with double radius of cross-section and half the length by $$1mm$$ is:

A
16J  B
8J  C
4J  D
14J  Solution

The correct option is C $$16J$$Stretching force, $$F=\cfrac { Y\pi { r }^{ 2 }\Delta L }{ L }$$where the symbols have their usual meanings.Both the wires are of same material, so $$Y$$ will be equal, extension in both the wires is same, so $$\Delta L$$ will be equal$$\therefore F\propto \cfrac { { r }^{ 2 } }{ L }$$$$\quad \therefore \cfrac { F' }{ F } =\cfrac { { (2r) }^{ 2 } }{ (L/2) } \times \cfrac { L }{ { r }^{ 2 } } =8$$or $$F'=8F...(i)\quad$$Work done is stretching a wire,$$W=\cfrac { 1 }{ 2 } \times F\times \Delta L\quad$$For sam extension$$W\propto F$$$$\therefore \cfrac { W' }{ W } =\cfrac { F' }{ F } =8$$ [Using (i)]$$W'=8W=8\times 2J=16J$$Physics

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