A sonometer wire is vibrating in resonance with a tuning fork. Keeping the tension applied same, the length of the wire is doubled. Under what conditions would the tuning fork still be is resonance with the wire ?

The transverse displacement of a string clamped at its both ends is given by 

$y\left( {x,t} \right) = 2\,\sin \,\left( {\frac{{2\pi }}{3}x} \right)\,\cos \,\left( {100\,\pi t} \right)$ 

where $x$ and $y$ are in $cm$ and $t$ is in $s$. Which of the following statements is correct ?

If the fundamental frequency of string is $220 \,cps$, the frequency of fifth harmonic will be ……… $cps$

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 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 ……… $Hz$