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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
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B
8J
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C
4J
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D
14J
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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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