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Question

One end of a long metallic wire of length $$L$$ area of cross-section $$A$$ and Young's modulus $$Y$$ is tied to the ceiling. The other end is tied to a massless spring of force constant $$k$$. A mass $$m$$ hangs freely from the free end of the spring. It is slightly pulled down and released. Its time period is given by-


A
2πmk
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B
2πmYAkL
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C
2πmkYA
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D
2πm(kL+YA)kYA
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Solution

The correct option is D $$\displaystyle 2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$$
$$F = \dfrac{YA\Delta l}{L} = k_2 \Delta l$$
we can consider the system as two springs in series hence 
$$\dfrac{1}{k_{eq}} = \dfrac{1}{k_1} +\dfrac{1}{k_2}$$
$$=\dfrac{1}{k} + \dfrac{L}{YA} = \dfrac{YAk +kL}{YAk}$$

$$ T = 2\pi \sqrt{\dfrac{m}{k_{eq}}} = 2\pi \sqrt{\dfrac{m(YAk + kL)}{YAk}}$$

Physics

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