A wire having a linear mass density
Question:

A wire having a linear mass density $9.0 \times 10^{-4} \mathrm{~kg} / \mathrm{m}$ is stretched between two rigid supports with a tension of $900 \mathrm{~N}$. The wire resonates at a frequency of $500 \mathrm{~Hz}$. The next higher frequency at which the same wire resonates is $550 \mathrm{~Hz}$. The length of the wire is_________m.

Solution:

$\mu=9.0 \times 10^{-4} \frac{\mathrm{kg}}{\mathrm{m}}$

$T=900 \mathrm{~N}$

$\mathrm{V}=\sqrt{\frac{\mathrm{T}}{\mu}}=\sqrt{\frac{900}{9 \times 10^{-4}}}=1000 \mathrm{~m} / \mathrm{s}$

$f_{1}=500 \mathrm{~Hz}$

$f=550$

$\frac{\mathrm{nV}}{2 \ell}=500$….(1)

$\frac{(\mathrm{n}+1) \mathrm{V}}{2 \ell}=500$……(2)

(ii) (i) $\frac{\mathrm{V}}{2 \ell}=50$

$\ell=\frac{1000}{2 \times 50}=10$