A wire having a linear mass density $9.0 \times 10^{-4} \;{kg} / {m}$ is stretched between two rigid supports with a tension of $900\; {N}$. The wire resonates at a frequency of $500\;{Hz}$. The next higher frequency at which the same wire resonates is $550\; {Hz}$. The length of the wire is $...... {m}$
JEE MAIN 2021, Diffcult
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$\mu=9.0 \times 10^{-4}\, \frac{{kg}}{{m}}$
${T}=900\, {N}$
${V}=\sqrt{\frac{{T}}{\mu}}=\sqrt{\frac{900}{9 \times 10^{-4}}}=1000 \,{m} / {s}$
${f}_{1}=500 \,{Hz}$
${f}=550$
$\frac{{nV}}{2 \ell}=500 \ldots . \text { (i) }$
$\frac{({n}+1) {V}}{2 \ell}=500 \ldots(ii)$
$(ii)\, (i)$ $\frac{{V}}{2 \ell}=50$
$\ell=\frac{1000}{2 \times 50}=10$
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