A stretched string of 1 m length, fixed at both ends, having a mass of 5 x ${10}^{-4}$ kg is under a tension of 20 N. It is plucked at a point situated at 25 cm from one end. The stretched string would vibrate with a frequency of-

Here, $\ell =1\mathrm{m},\mathrm{m}=5\times {10}^{-4}\mathrm{kg},\mathrm{T}=20\mathrm{N}$
We have, $v=\frac{1}{2\ell }\sqrt{\frac{T}{\mu }}=\frac{1}{2\left(1\right)}\sqrt{\frac{20}{5\times {10}^{-4}}}=\frac{1}{2}\sqrt{4\times {10}^4}=\frac{2\times {10}^2}{2}={10}^2=100\mathrm{Hz}$

if it is plucked at 25 cm 

$$\begin{gathered} \frac{\mathrm{\lambda }}{4}=25 \mathrm{cm}, \mathrm{\lambda }=100 \mathrm{cm} \mathrm{fundamental} \mathrm{mode} \mathrm{\lambda }=2\mathrm{l}=200\mathrm{cm} \\ \mathrm{first} \mathrm{overtone} \mathrm{or} \mathrm{second} \mathrm{harmonic} : 2\mathrm{l}/2=100\mathrm{cm} \mathrm{so} \mathrm{it} \mathrm{is} \mathrm{vibrating} \mathrm{in} \mathrm{the} \\ \mathrm{second} \mathrm{harmonic} \mathrm{i}.\mathrm{e}. 200 \mathrm{Hz} \end{gathered}$$