One end of a long metallic wire of length $L$ is tied to the ceiling. The other end is tied to massless spring of spring constant $K$. A mass $ m$ hangs freely from the free end of the spring. The area of cross-section and Young's modulus of the wire are $A$ and $Y$ respectively. If the mass is slightly pulled down and released, it will oscillate with a time period $T$ equal to
IIT 1993, Diffcult
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(b) The wire may be treated as a string for which force constant
${k_1} = \frac{{Force}}{{Extension}} = \frac{{YA}}{L}\;\;\;\left( {\because Y = \frac{F}{A} \times \frac{L}{{\Delta L}}} \right)$
Spring constant of the spring ${k_2} = K$
Hence spring constant of the combination (series) ${k_{eq}} = \frac{{{k_1}{k_2}}}{{{k_1} + {k_2}}} $
$= \frac{{(YA/L)K}}{{(YA/L) + K}}$
$= \frac{{YAK}}{{YA + KL}}$
Time period $T = 2\pi \sqrt {\frac{m}{k}} = 2\pi {\left[ {\frac{{(YA + KL)m}}{{YAK}}} \right]^{1/2}}$
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