$A$ wire of length $0.3\,m$,stretched between rigid supports,has its $n^{\text{th}}$ and $(n+1)^{\text{th}}$ harmonics at $400\,Hz$ and $450\,Hz$,respectively. If the tension in the string is $2700\,N$,its linear mass density is ......... $kg/m$.

The frequency of transverse vibrations in a stretched string is $200 \ Hz$. If the tension is increased four times and the length is reduced to one-fourth the original value,the frequency of vibration will be .... $Hz$.

The fundamental frequency of a sonometer wire increases by $9 \ Hz$,if its tension is increased by $69 \%$,keeping the length constant. The frequency of the wire is (in $Hz$)

The first overtone of a stretched wire of given length is $320 \ Hz$. The first harmonic is .... $Hz$.

$A$ steel wire of length $1.25 \ m$ is stretched between two rigid supports. The tension in the wire produces an elastic strain of $0.14 \%$. The fundamental frequency of the wire is (Density and Young's modulus of steel are $7.7 \times 10^3 \ kg \ m^{-3}$ and $2.2 \times 10^{11} \ N \ m^{-2}$ respectively). (in $Hz$)

To increase the frequency of a stretched string from $100 \ Hz$ to $400 \ Hz$,the tension in the string has to be changed by how many times?