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

Two vibrating strings of the same material but lengths $L$ and $2 L$ have radii $2 r$ 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 $n_{1}$ and the other with frequency $n_{2}$. The ratio $\frac{n_1}{n_2}$ is given by

Two uniform wires of the same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and radius of the first wire is twice the radius of the second wire then the ratio of the lengths of the first wire to second wire is

If the tension and diameter of a sonometer wire of fundamental frequency n are doubled and density is halved, then its fundamental frequency will become 

When the length of the vibrating segment of a sonometer wire is increased by $1 \%,$ the percentage change in its frequency is

A $20 \mathrm{cm}$ long string having a mass of $1.0 \mathrm{g}$ is fixed at both the ends. The tension in the string is $0.5 \mathrm{N}$. The string is set into vibrations using an external vibrator of frequency $100 \mathrm{Hz}$. Find the separation (in $\mathrm{cm}$ ) between the successive nodes on the string. 

When a string is divided into three segments of lengths$l_1, l_2 \text{and} l_3$  the fundamental frequencies of these three segments are ${\nu }_1, {\nu }_2 \text{and} {\nu }_3$ respectively. The original fundamental frequency $\nu$ of the string is 

If $n_1, n_2, n_3$ are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency n of the string is given by