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…..

Q: 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) $\displaystyle 2\pi \sqrt{\frac{m}{k}}$

(b) $\displaystyle 2\pi \sqrt{\frac{mYA}{k L}}$

(c) $\displaystyle 2\pi \sqrt{\frac{mY}{k}}$

(d) $\displaystyle 2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$

Ans:(d)

Sol: For wire ,

$\displaystyle Y = \frac{F L}{A \Delta L}$

$\displaystyle F = (\frac{A Y}{L} )\Delta L$

$\displaystyle F = K’ \Delta L$

Where $\displaystyle K’ = \frac{A Y}{L}$

For Spring , Force Constant = K

Both Wire & Spring are in Series ,

$\displaystyle \frac{1}{K_{eq}} = \frac{1}{K} + \frac{1}{K’}$

$\displaystyle \frac{1}{K_{eq}} = \frac{1}{K} + \frac{L}{A Y}$

Time Period , $\displaystyle T = 2\pi \sqrt{\frac{m}{K_{eq}}}$

$\displaystyle T = 2\pi \sqrt{m(\frac{1}{K} + \frac{L}{A Y})}$

$\displaystyle T = 2\pi \sqrt{\frac{m(YA + KL)}{kYA}}$