$A$ sonometer wire resonates with $4$ antinodes between the two bridges for a given tuning fork when a $1 \text{ kg}$ mass is suspended from the wire. Using the same fork, when mass $M$ is suspended, the wire resonates producing $2$ antinodes between the two bridges. (The distance between the bridges remains the same). The value of $M$ is: (in $\text{ kg}$)

$A$ wire of length $L$ and mass per unit length $6.0 \times 10^{-3} \; kg/m$ is put under a tension of $540 \; N$. Two consecutive frequencies at which it resonates are $420 \; Hz$ and $490 \; Hz$. Then $L$ in meters is: (in $; m$)

$A$ sonometer wire resonates with $4$ antinodes between two bridges for a given tuning fork, when $1 \,kg$ mass is suspended from the wire. Using the same fork, when mass $M$ is suspended, the wire resonates producing $2$ antinodes between the two bridges (the distance between the two bridges remains the same). The value of $M$ is: (in $\,kg$)

An iron load of $2 \ kg$ is suspended in air from the free end of a sonometer wire of length $1 \ m$. $A$ tuning fork of frequency $256 \ Hz$ is in resonance with $\frac{1}{\sqrt{7}}$ times the length of the sonometer wire. If the load is immersed in water, the length of the wire in metres that will be in resonance with the same tuning fork is (Specific gravity of iron $= 8$).

$A$ sonometer wire of resonating length $90 \ cm$ has a fundamental frequency of $400 \ Hz$ when kept under some tension. The resonating length of the wire with a fundamental frequency of $600 \ Hz$ under the same tension is . . . . . . $cm$.

Two wires are fixed in a sonometer. Their tensions are in the ratio $8:1$. The lengths are in the ratio $36:35$. The diameters are in the ratio $4:1$. Densities of the materials are in the ratio $1:2$. If the lower frequency in the setting is $360 \ Hz$,the beat frequency when the two wires are sounded together is: (in $Hz$)