The frequency of a sonometer wire is $100 \ Hz$. When the weights producing the tension are completely immersed in water,the frequency becomes $80 \ Hz$,and on immersing the weights in a certain liquid,the frequency becomes $60 \ Hz$. The specific gravity of the liquid 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 ............ $Hz$.

Two vibrating strings $A$ and $B$ of the same material but lengths $3 L$ and $2 L$ have radii $2 r$ and $3 r$ respectively. They are stretched under the same tension. String $A$ vibrates in the second overtone and string $B$ in the fundamental mode. The ratio of their frequencies $n_A / n_B$ will be:

$A$ string in a musical instrument is $50 \ cm$ long and its fundamental frequency is $800 \ Hz$. If a frequency of $1000 \ Hz$ is to be produced,then the required length of the string is ..... $cm$.

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:

To increase the frequency of a stretched string from $100 \ Hz$ to $400 \ Hz$,the tension in the string has to be changed by how many times?