$A$ sonometer wire is in unison with a tuning fork when it is stretched by weight $w$ and the corresponding resonating length is $L_1$. If the weight is reduced to $\frac{w}{4}$,the corresponding resonating length becomes $L_2$. The ratio $\frac{L_1}{L_2}$ is:

The fundamental frequency of a sonometer wire of length $l$ is $n_0$. $A$ bridge is now introduced at a distance of $\Delta l$ (where $\Delta l \ll l$) from the center of the wire. The lengths of wire on the two sides of the bridge are now vibrated in their fundamental modes. Then,the beat frequency is nearly

$A$ tuning fork gives $4$ beats with $50 \ cm$ length of a sonometer wire. If the length of the wire is shortened by $1 \ cm$,the number of beats is still the same. The frequency of the fork is (in $Hz$)

$A$ wire of density $8 \times 10^3\,kg/m^3$ is stretched between two clamps $0.5\,m$ apart. The extension developed in the wire is $3.2 \times 10^{-4}\,m$. If Young's modulus $Y = 8 \times 10^{10}\,N/m^2$,the fundamental frequency of vibration in the wire will be $......\,Hz$.

$A$ 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 $; m$)

$A$ $1 \, cm$ long string vibrates with a fundamental frequency of $256 \, Hz$. If the length is reduced to $\frac{1}{4} \, cm$ keeping the tension unaltered,the new fundamental frequency will be: (in $, Hz$)