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Question

Two wires $$A$$ and $$B$$ have equal lengths and are made of the same material, but the diameter of wire $$A$$ is twice that of wire $$B$$. Then, for a given load:


A
the extension of B will be four times that of A
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B
the extensions of A and B will be equal
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C
the strain in B is four times that in A
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D
the strains in A and B will be equal
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Solution

The correct options are
A the extension of $$B$$ will be four times that of $$A$$
C the strain in $$B$$ is four times that in $$A$$
$$Y=\dfrac{F/\pi \left ( \dfrac{d}{2} \right )^2}{\Delta l/l}$$
$$Y=\dfrac{F/\pi \left ( \dfrac{d_A}{2} \right )^2}{\Delta l_A/l}$$-----(1)
$$Y=\dfrac{F/\pi \left ( \dfrac{d_B}{2} \right )^2}{\Delta l_B/l}$$-----(2)
Dividing (1) by (2)
$$1=\dfrac{\pi \left ( \dfrac{d_B}{2} \right )^2}{\pi \left ( \dfrac{d_A}{2} \right )^2}\times \dfrac{\Delta l_B}{\Delta l_A}$$
$$1=\left ( \dfrac{d_B}{d_A} \right )^2\times \dfrac{\Delta l_B}{\Delta l_A}$$
$$1=\left ( \dfrac{1}{2} \right )^2\times \dfrac{\Delta l_B}{\Delta l_A}$$
$$\dfrac{\Delta l_A}{\Delta l_B}=\dfrac{1}{4}$$
strain$$=\dfrac{\Delta l}{l}$$
So, extension in B will be four times of A and strain in B will be four times of A since $$\Delta_B=4\Delta_A$$ and length of A and B are same.

Physics

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