A sonometer wire of resonating length $90 \mathrm{~cm}$ has a fundamental frequency of $400 \mathrm{~Hz}$ when kept under some tension. The resonating length of the wire with fundamental frequency of $600 \mathrm{~Hz}$ under same tension________.$\mathrm{cm}$.
JEE MAIN 2024, Diffcult
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$\mathrm{f}_0=400 \mathrm{~Hz} ; \mathrm{v}=\sqrt{\frac{\mathrm{T}}{\mu}}=\text { constant }$
$\frac{\lambda}{2}=\mathrm{L} ; \mathrm{v}=\mathrm{f}_0 \lambda$
$\frac{\mathrm{v}}{2 \mathrm{f}_0}=\mathrm{L} \Rightarrow \mathrm{v}=2 \mathrm{Lf}_0$
$\mathrm{~L}^{\prime}=\frac{\mathrm{v}}{2 \mathrm{f}^{\prime}}=\frac{2 \mathrm{Lf}_0}{2 \mathrm{f}^{\prime}}$
$=\frac{L f_0}{\mathrm{f}^{\prime}}=\frac{90 \times 400}{600}=60$
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