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

Answer 1

Answer: (D)

For oscillating mass at end of a rod. Restoring force
$=\frac{YA}{L}\cdot x$
So, $k_1=$ spring constant for a rod is $\frac{YA}{L}$.
If a rod and spring are connected, then it is a series combination.

So, $\left(k_{eq}\right)$ system $=\frac{k_1{k}_2}{k_1+k_2}$
$=\frac{kYA/L}{k+\frac{YA}{L}}=\frac{kYA}{kL+YA}$
So, $T=2\pi \sqrt{\frac{m}{k_{eq}}}$
$\Rightarrow T=2\pi \sqrt{\frac{m(kL+YA)}{kYA}}$