The length of the wire shown in the figure between the pulleys is $1.5 \, m$ and its mass is $12.0 \, g$. The frequency of vibration with which the wire vibrates in three loops,forming an antinode at the midpoint of the wire,is: (Given $g = 9.8 \, m/s^2$)

$A$ wire of length $1 \,m$ under a certain initial tension emits a sound of fundamental frequency $256 \,Hz$. When the tension is increased by $1 \,kg \,wt$,the frequency of the fundamental note increases to $320 \,Hz$. The initial tension is ........... $kg \,wt$.

Two similar sonometer wires have fundamental frequencies of $500 \, Hz$. They are under the same tension. By what percentage should the tension be increased in one wire so that the two wires produce $5 \, \text{beats/sec}$ (in $\%$)?

Two strings $A$ and $B$ of lengths $L_A = 80 \text{ cm}$ and $L_B = x \text{ cm}$ respectively are used separately in a sonometer. The ratio of their densities $(d_A / d_B)$ is $0.81$. The diameter of $B$ is one-half that of $A$. If the strings have the same tension and fundamental frequency,the value of $x$ is:

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 length $L$,diameter $d$,and density of material $\rho$ is under tension $T$,having a fundamental frequency of vibration $n_A$. Another wire of length $2L$,tension $2T$,density $2\rho$,and diameter $3d$ has a fundamental frequency of vibration $n_B$. The ratio $n_B : n_A$ is