$A$ stretched wire of length $110 \, cm$ is divided into three segments whose frequencies are in the ratio $1 : 2 : 3$. Their lengths must be:

The fundamental frequency of a sonometer wire increases by $6\, Hz$ if its tension is increased by $44\%$,keeping the length constant. The frequency of the wire is ...... $Hz$.

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

$A$ wire stretched between two rigid supports vibrates in its fundamental mode with a frequency of $45 \;Hz$. The mass of the wire is $3.5 \times 10^{-2} \;kg$ and its linear mass density is $4.0 \times 10^{-2} \;kg \;m^{-1}$. What is
$(a)$ the speed of a transverse wave on the string,and
$(b)$ the tension in the string?

When a string of length $l$ is divided into three segments of length $l_1, l_2$ and $l_3$,the fundamental frequencies of the three segments are $n_1, n_2$ and $n_3$ respectively. The original fundamental frequency $n$ of the string is:

The length of a sonometer wire is $0.75\, m$ and density is $9 \times 10^3\, kg/m^3$. It can bear a stress of $8.1 \times 10^8\, N/m^2$ without exceeding the elastic limit. What is the fundamental frequency that can be produced in the wire in $Hz$?