A string of length $1 \mathrm{~m}$ and mass $2 \times 10^{-5} \mathrm{~kg}$ is under tension $\mathrm{T}$. when the string vibrates, two successive harmonics are found to occur at frequencies $750 \mathrm{~Hz}$ and $1000 \mathrm{~Hz}$. The value of tension $\mathrm{T}$ is. . . . . . .Newton.
IIT 2023, Medium
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$\mathrm{f}=\frac{\mathrm{P}}{2 \ell} \sqrt{\frac{\mathrm{T}}{\mu}}$
$750=\frac{\mathrm{P}}{2} \sqrt{\frac{\mathrm{T}}{\mu}} \ldots . .(1)$
$1000=\frac{\mathrm{P}+1}{2} \sqrt{\frac{\mathrm{T}}{\mu}} \ldots . .(2)$
$\frac{4}{3}=\frac{\mathrm{P}+1}{\mathrm{P}} \quad \therefore \mathrm{P}=3$
$\Rightarrow 1000=\frac{4}{2} \sqrt{\frac{\mathrm{T}}{2 \times 10^{-5}}} \therefore \mathrm{T}=5 \mathrm{~N}$
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