A string is fixed at both ends vibrates in a resonant mode with a separation $2.0 \,\,cm$ between the consecutive nodes. For the next higher resonant frequency, this separation is reduced to $1.6\,\, cm$. The length of the string is …. $cm$

Two similar sonometer wires given fundamental frequencies of $500Hz$. These have same tensions. By what amount the tension be increased in one wire so that the two wires produce $5$ beats/sec …. $\%$

A wire of length $L$ and mass per unit length $6.0\times 10^{-3}\; \mathrm{kgm}^{-1}$ is put under tension of $540\; \mathrm{N}$. Two consecutive frequencies that it resonates at are : $420\; \mathrm{Hz}$ and $490 \;\mathrm{Hz}$. Then $\mathrm{L}$ in meters is

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

A second harmonic has to be generated in a string of length $l$ stretched between two rigid supports. The point where the string has to be plucked and touched are