Question

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

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

Answer 1

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

young's modulus $Y=\frac{F\times L}{A\times △L}$ or $\frac{F}{△L}=\frac{YA}{L}$
Force constant of wire $k_1=\frac{F}{△L}=\frac{YA}{L}$
Force constant of wire $k_1$ and spring of force constant $k$ are in series
$\frac{1}{k_2}=\frac{1}{k}+\frac{1}{k_1}$
or $k_2=\frac{k{k}_1}{k+k_1}=\frac{k\left(\frac{YA}{L}\right)}{k+\frac{YA}{L}}=\frac{kYA}{kL+YA}$
The time period of the combination is
$T^{\prime }=2\pi \sqrt{\frac{m}{k_2}}=2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$.