$A$ string is divided into three segments such that the segments possess fundamental frequencies in the ratio $1: 2: 3$. Then,the lengths of the segments are in the ratio:

An iron load of $2 \ kg$ is suspended in air from the free end of a sonometer wire of length $1 \ m$. $A$ tuning fork of frequency $256 \ Hz$ is in resonance with $\frac{1}{\sqrt{7}}$ times the length of the sonometer wire. If the load is immersed in water,the length of the wire in metres that will be in resonance with the same tuning fork is (Specific gravity of iron $= 8$).

$A$ sonometer wire,with a suspended mass of $M = 1 \, kg$,is in resonance with a given tuning fork. The apparatus is taken to the moon where the acceleration due to gravity is $(1/6)$ that on earth. To obtain resonance on the moon,the value of $M$ should be ......... $kg$.

The fundamental frequency of a string stretched with a weight of $4 \ kg$ is $256 \ Hz$. The weight required to produce its octave is .... $kg \ wt$.

$A$ wire of density $8 \times 10^3\,kg/m^3$ is stretched between two clamps $0.5\,m$ apart. The extension developed in the wire is $3.2 \times 10^{-4}\,m$. If Young's modulus $Y = 8 \times 10^{10}\,N/m^2$,the fundamental frequency of vibration in the wire will be $......\,Hz$.

At the poles,a stretched wire of a given length vibrates in unison with a tuning fork. At the equator,for the same setting to produce resonance with the same fork,the vibrating length of the wire