Solution

Given,

The length of the wire (l) = 2m
Mass of the wire (M) = 3g = 3 x 10^-3 Kg

∴ The mass per unit length, \(m=\frac{3\times10^{-3}}2\) kg/m

Tension (T) = 500 N
Frequency of fundamental mode of vibration (f) = ?
The wavelength of the fundamental mode of vibration (λ) = ?

Now, for fundamental mode of vibration, we have,

\(f=\frac1{2l}\sqrt{\frac Tm}=\frac1{2\times2}\sqrt{\frac{\displaystyle\frac{500}{3\times10^{-3}}}2}=\frac14\times\sqrt{\frac{1000}{3\times10^{-3}}}=\frac14\sqrt{\frac{10^6}3}\)
∴ f = 144.3 Hz

Also, in this case, we have

\(\frac\lambda2=l\)
Or, \(\lambda=2l=2\times2=4\) m

Hence, the required values of frequency and wavelength are 144.3 Hz and 4 m respectively.