$A$ tuning fork resonates with a sonometer wire of length $1 \ m$ stretched with a tension of $6 \ N$. When the tension in the wire is changed to $54 \ N$,the same tuning fork produces $12$ beats per second with it. The frequency of the tuning fork is $Hz$.

When a vibrating tuning fork is placed on a sound box of a sonometer,$8$ beats per second are heard when the length of the sonometer wire is kept at $101 \,cm$ or $100 \,cm$. Then the frequency of the tuning fork is (Consider that the tension in the wire is kept constant). (in $\,Hz$)

$A$ string is clamped at both ends and it is vibrating in its $4^{th}$ harmonic. The equation of the stationary wave is $Y = 0.3 \sin(0.157 x) \cos(200\pi t)$. The length of the string is ..... $m$ (all quantities are in $SI$ units).

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

In an experiment with a sonometer,when a mass of $180\,g$ is attached to the string,it vibrates with a fundamental frequency of $30\,Hz$. When a mass $m$ is attached,the string vibrates with a fundamental frequency of $50\,Hz$. The value of $m$ is $.........\,g$.

$A$ string with a mass density of $4 \times 10^{-3} \, kg/m$ is under a tension of $360 \, N$ and is fixed at both ends. One of its resonance frequencies is $375 \, Hz$. The next higher resonance frequency is $450 \, Hz$. The mass of the string is: