A string of length $1\,\,m$ and linear mass density $0.01\,\,kgm^{-1}$ is stretched to a tension of $100\,\,N.$ When both ends of the string are fixed, the three lowest frequencies for standing wave are $f_1, f_2$ and $f_3$. When only one end of the string is fixed, the three lowest frequencies for standing wave are $n_1, n_2$ and $n_3$. Then

A tuning fork gives $4$ beats with $50\, cm$ length of a sonometer wire if the length of  the wire is shortened by $1\, cm$. the no. of beats still the same. The frequency of the  fork is -............. $\mathrm{Hz}$

A wire having a linear mass density $9.0 \times 10^{-4} \;{kg} / {m}$ is stretched between two rigid supports with a tension of $900\; {N}$. The wire resonates at a frequency of $500\;{Hz}$. The next higher frequency at which the same wire resonates is $550\; {Hz}$. The length of the wire is $...... {m}$

Transverse waves of same frequency are generated in two steel wires $A$ and $B$. The diameter of $A$ is twice of $B$ and the tension in $A$ is half that in $B$. The ratio of velocities of wave in $A$ and $B$ is

The length of the wire shown in 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 antinode at the mid point of the wire is $(g = 9.8 \,m/s^2)$

A uniform metal wire of density $\rho $, cross-sectional area $A$ and length $L$ is stretched with a tension $T$. The speed of transverse wave in the wire is given by