The correct graph between the frequency $n$ and the square root of density $(\rho)$ of a wire,keeping its length,radius,and tension constant,is:

The speed of a transverse wave on a string is $160 \,m/s$. If the three resonant frequencies of this string are $160 \,Hz$, $240 \,Hz$, and $400 \,Hz$ respectively, the length of the string is: (in $\,cm$)

In a sonometer wire,the tension is maintained by suspending a $50.7 \, kg$ mass from the free end of the wire. The suspended mass has a volume of $0.0075 \, m^3$. The fundamental frequency of the wire is $260 \, Hz$. If the suspended mass is completely submerged in water,the fundamental frequency will become .... $Hz$ (take $g = 10 \, ms^{-2}$)

If the length of a stretched string is reduced by $40 \%$ and the tension is increased by $44 \%$,then the ratio of the final to the initial frequencies of the stretched string is:

$A$ piano string $1.5 \, m$ long is made of steel of density $7.7 \times 10^3 \, kg/m^3$ and Young's modulus $2 \times 10^{11} \, N/m^2$. It is maintained at a tension which produces an elastic strain of $1 \%$ in the string. The fundamental frequency of transverse vibrations of the string is ......... $Hz$.

When tension $T$ is applied to a sonometer wire of length $\ell$,it vibrates with the fundamental frequency $n$. Keeping the experimental setup same,when the tension is increased by $8 \ N$,the fundamental frequency becomes three times the earlier fundamental frequency $(n)$. The initial tension applied to the wire in newton was: