$A$ string of length $100 \,cm$ has three resonant frequencies, $120 \,Hz, 200 \,Hz$ and $280 \,Hz$. If a node is formed at the end of the string, the speed of the transverse wave on this string is : (in $\,m/s$)

$A$ string fixed at both ends is vibrating in two segments. The wavelength of the corresponding wave is

$A$ wire of length $0.3\,m$,stretched between rigid supports,has its $n^{\text{th}}$ and $(n+1)^{\text{th}}$ harmonics at $400\,Hz$ and $450\,Hz$,respectively. If the tension in the string is $2700\,N$,its linear mass density is ......... $kg/m$.

If the length of a stretched string is shortened by $40 \%$ and the tension is increased by $44 \%$,then the ratio of the final and initial fundamental frequencies is :

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 $2^{\text{nd}}$ overtone of the $2^{\text{nd}}$ wire and the radius of the first wire is twice the radius of the $2^{\text{nd}}$ wire,then the ratio of the length of the first wire to the $2^{\text{nd}}$ wire is:

$A$ stretched string is vibrating in the second overtone. The number of nodes and antinodes between the ends of the string are respectively: