If the length of a stretched string is shortened by $40\%$ and the tension is increased by $44\%$,then the ratio of the final and initial fundamental frequencies is

The equation of vibration of a stretched string fixed at both ends and vibrating in $5^{\text{th}}$ harmonic is $Y = 3 \sin(0.4x) \cos(200\pi t)$, where '$x$' and '$Y$' are in $cm$ and '$t$' is in seconds. The length of the string is: (in $\pi \text{ cm}$)

$A$ sonometer wire resonates with a given tuning fork,forming a standing wave with $5$ antinodes between two bridges when a mass of $9 \ kg$ is suspended from the wire. When a mass '$m$' is suspended from the wire,with the same tuning fork and the same length between the two bridges,$3$ antinodes are formed. The value of mass '$m$' is: (in $kg$)

$A$ string of length $1 \, m$ fixed at both ends is vibrating in the $3^{rd}$ overtone. The tension in the string is $200 \, N$ and the linear mass density is $5 \, g/m$. The frequency of these vibrations is ..... $Hz$.

$A$ device used for investigating the vibration of a fixed string or wire is

Calculate the frequency of the second harmonic formed on a string of length $0.5 \ m$ and mass $2 \times 10^{-4} \ kg$ when stretched with a tension of $20 \ N$.