$A$ wire of length $L$ and mass per unit length $6.0 \times 10^{-3} \; kg/m$ is put under a tension of $540 \; N$. Two consecutive frequencies at which it resonates are $420 \; Hz$ and $490 \; Hz$. Then $L$ in meters is: (in $; m$)

Two identical strings of length $\ell$ and $2\ell$ vibrate with fundamental frequencies $N$ Hz and $1.5N$ Hz,respectively. The ratio of tensions for the smaller length to the larger length is

The fundamental frequency of a wire stretched by $2 \ kgwt$ is $100 \ Hz$. The weight required to produce its octave is (in $kgwt$)

$A$ 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 $Hz$)

Two strings $A$ and $B$ of lengths $L_A = 80 \text{ cm}$ and $L_B = x \text{ cm}$ respectively are used separately in a sonometer. The ratio of their densities $(d_A / d_B)$ is $0.81$. The diameter of $B$ is one-half that of $A$. If the strings have the same tension and fundamental frequency,the value of $x$ is:

Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$,the ratio of the length of string $B$ to that of $A$ is