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. $2\pi \sqrt{\frac{m}{k}}$
• B. $2\pi \sqrt{\frac{mYA}{kL}}$
• C. $2\pi \sqrt{\frac{mk}{YA}}$
• D. $2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$

Answer 1

The correct option is D $2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$

$F=\frac{YA\Delta l}{L}=k_2\Delta l$

we can consider the system as two springs in series hence

$\frac{1}{k_{eq}}=\frac{1}{k_1}+\frac{1}{k_2}$

$=\frac{1}{k}+\frac{L}{YA}=\frac{YAk+kL}{YAk}$

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