Three similar wires of frequencies $n_1, n_2$ and $n_3$ are joined to form a single wire. What will be its resulting frequency $n$?

The frequency of a sonometer wire is $f$. When the weights producing the tension are completely immersed in water,the frequency becomes $f/2$. When the weights are immersed in a certain liquid,the frequency becomes $f/3$. The specific gravity of the liquid is:

$A$ string is stretched between two fixed points separated by $75\,cm$. It is observed to have resonant frequencies of $420\,Hz$ and $315\,Hz$. There are no other resonant frequencies between these two. Then,the lowest resonant frequency for this string is ..... $Hz$.

Two vibrating strings of the same material but lengths $L$ and $2L$ have radii $2r$ and $r$ respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes,the one of length $L$ with frequency $f_1$ and the other with frequency $f_2$. The ratio $\frac{f_1}{f_2}$ is given by

$A$ steel wire of length $1 \,m$, mass $0.1 \,kg$, and uniform area of cross-section $10^{-6} \,m^2$ is rigidly fixed at both ends without any tension. Its temperature is lowered by $20^{\circ} C$ and transverse waves are set up by plucking the wire at the middle. The frequency of the fundamental mode is ($Y = 200 \,GPa$, $\alpha = 1.21 \times 10^{-5} {}^{\circ} C^{-1}$). (in $\,Hz$)

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 $\,cm$)