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 string is ……… $Hz$

The transverse displacement in a streched string is given by
$y = 0.06 \sin \, \left( {\frac{{2\pi }}{3}x} \right)\cos \,(120\pi t)$

where $x$ and $y$ are in $m$ and $t$ is in $s$. The length of the string is $1.5\, m$ and  its mass is $3.0 \times 10^{-2} \,kg$, then tension in string is ….. $N$

A tuning fork of frequency $340\, Hz$ is sounded above an organ pipe of length $120\, cm$. Water is now slowly poured in it. The minimum height of water column required for resonance is …. $cm$ (speed of sound in air $= 340 \,m/s$)

To increase the frequency from $100 Hz$ to $400 Hz$ the tension in the string has to be changed by ….. $times$

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