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 $k$ and a mass $m$ is hung from the free end of the spring. If $m$ is slightly pulled down and released, then its time period of oscillation is

• A

  $2 \pi \sqrt{\frac{m}{k}}$
• B

  $2 \pi \sqrt{\frac{m Y A}{k L}}$
• C

  $2 \pi \sqrt{\frac{m(k A+Y L)}{k Y A}}$
• Correct answer
  D

  $2 \pi \sqrt{\frac{m(k L+Y A)}{k Y A}}$

Solution

For oscillating mass at end of a rod. Restoring force $$ =\frac{Y A}{L} \cdot x $$
So, $k_1=$ spring constant for a rod is $\frac{Y A}{L}$. If a rod and spring are connected, then it is a series combination. So, $\left(k_{\text {eq }}\right)$ $$ \begin{aligned} \text { system } & =\frac{k_1 k_2}{k_1+k_2} \\ & =\frac{k Y A / L}{k+\frac{Y A}{L}}=\frac{k Y A}{k L+Y A} \end{aligned} $$ So, $\quad T=2 \pi \sqrt{\frac{m}{k_{e q}}} \Rightarrow T=2 \pi \sqrt{\frac{m(k L+Y A)}{k Y A}}$

Hence, the correct answer is option: 'D'.