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

The fundamental frequency of a sonometer with a weight of $4\,kg$ is $256\,Hz$ . The weight required to produce its octave is …. $kg-wt$

The first overtone of a stretched wire of given length is $320 Hz$. The first harmonic is  …. $Hz$

A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $9 kg$ is suspended from the wire. When this mass is replaced by a mass $M$, the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of $M$ is … $kg$

A string is stretched so that its length is increased by $\frac{1}{\eta }$ of its original length. The ratio of fundamental frequency of transverse vibration to that of fundamental frequency of longitudinal vibration will be