Question:
Two identical strings $X$ and $Z$ made of same material have tension $T_{X}$ and $T_{Z}$ in them. If their fundamental frequencies are $450 \mathrm{~Hz}$ and $300 \mathrm{~Hz}$, respectively, then the ratio $T_{X} / T_{Z}$ is:
Correct Option: 1
Solution:
(1) Using $f=\frac{1}{2 \ell} \sqrt{\frac{T}{\mu}}$,
where, $T=$ tension and $\mu=\frac{\text { mass }}{\text { length }}$
$f_{x}=\frac{1}{2 \ell} \sqrt{\frac{T_{x}}{\mu}}$ and $f_{z}=\frac{1}{2 \ell} \sqrt{\frac{T_{z}}{\mu}}$
$\frac{f_{x}}{f_{z}}=\frac{450}{300}=\sqrt{\frac{T_{x}}{T_{z}}}$
$\therefore \frac{T_{x}}{T_{z}}=\frac{9}{4}=2.25$
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