The fundamental frequency of a sonometer wire is increases by $6\, Hz$ if its tension is increased by $44\%$, keeping the length constant. The frequency of the wire is …… $Hz$

A string is stretched between fixed points separated by $75.0\ cm$. It is observed to have resonant frequencies of $420\ Hz$ and $315\ Hz.$ There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string is  …. $Hz$

The pattern of standing waves formed on a stretched string at two instants of time are shown in figure. The velocity of two waves superimposing to form stationary waves is $360$ $\mathrm{ms}^{-1}$ and their frequencies are $256$ $\mathrm{Hz}$.

$(a)$ Calculate the time at which the second curve is plotted.

$(b)$ Mark nodes and antinodes on the curve.

$(c)$ Calculate the distance between $\mathrm{A}^{\prime}$ and $\mathrm{C}^{\prime}$.

A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $9\,kg$ is suspended from the wire. When this mass is replaced by a mass $M,$ the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of $M$ is …. $kg$

A string of length $2 m$ is fixed at both ends. If this string vibrates in its fourth normal mode with a frequency of $500 Hz$ then the waves would travel on its with a velocity of ….. $m/s$