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  4. 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

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Q. 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

AP EAMCETAP EAMCET 2018

Solution:

image
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.
image
So, $\left(k_{ eq }\right)$ system $=\frac{k_{1} k_{2}}{k_{1}+k_{2}}$
$=\frac{k Y A / L}{k+\frac{Y A}{L}}=\frac{k Y A}{k L+Y A}$
So, $T=2 \pi \sqrt{\frac{m}{k_{e q}}}$
$\Rightarrow T=2 \pi \sqrt{\frac{m(k L+Y A)}{k Y A}}$