Question

One end of a long metallic wire of length $L$ is tied to the ceiling. The other end is tied to a 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
[Assume that , within the elastic limit, wire behaves as a spring]

• A. $2\pi \left(\frac{m}{k}\right)$
• B. $2\pi \sqrt{\frac{(YA+kl)m}{YAk}}$
• C. $2\pi \frac{mYA}{kL}$
• D. $2\pi \frac{mL}{YA}$

Answer 1

The correct option is B $2\pi \sqrt{\frac{(YA+kl)m}{YAk}}$

The wire can be treated as a spring with force constant
$k_1=\frac{YA}{l}$
and Force constant of spring, $k_2=k$ [given]
$k_{\text{eq}}=\frac{k_1{k}_2}{k_1+k_2}=\frac{\left(\frac{YA}{L}\right)\left(k\right)}{\frac{YA}{L}+k}=\frac{YAk}{YA+kL}$
Time period of oscillation
$T=2\pi \sqrt{\frac{m}{k_{eq}}}=2\pi \sqrt{\frac{(YA+kL)m}{YAk}}$
Thus, option (b) is the correct answer.