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 lengths of the strings is :

The fundamental frequency of a sonometer wire is $n$. If the tension is increased $3$ times,the length is increased $3$ times,and the diameter is increased $2$ times,what will be the new frequency?

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

$A$ tuning fork vibrating with a sonometer having a wire of length $20 \,cm$ produces $5$ beats per second. The beat frequency does not change if the length of the wire is changed to $21 \,cm$. The frequency of the tuning fork must be ............ $Hz$.

$A$ stationary wave is produced along a stretched string of length $80 \,cm$. The resonant frequencies of the string are $90 \,Hz$, $150 \,Hz$, and $210 \,Hz$. The speed of the transverse wave in the string is: (in $\,m/s$)

$A$ 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 $\,Hz$)