The length of the string of a musical instrument is $90 \;cm$ and has a fundamental frequency of $120 \;Hz$. Where (in $cm$) should it be pressed to produce a fundamental frequency of $180 \;Hz$?

$A$ stretched string resonates with a tuning fork of frequency $512 \; Hz$ when the length of the string is $0.5 \; m$. The length of the string required to vibrate resonantly with a tuning fork of frequency $256 \; Hz$ would be .......... $m$.

$A$ tuning fork gives $5 \text{ beats per second}$ with a $33 \text{ cm}$ length of sonometer wire. If the length of the wire is shortened by $1 \text{ cm}$, the number of beats is still the same. The frequency of the fork is (in $\text{ Hz}$)

Two wires $W_1$ and $W_2$ have the same radius $r$ and respective densities $\rho_1$ and $\rho_2$ such that $\rho_2 = 4\rho_1$. They are joined together at the point $O$,as shown in the figure. The combination is used as a sonometer wire and kept under tension $T$. The point $O$ is midway between the two bridges. When a stationary wave is set up in the composite wire,the joint is found to be a node. The ratio of the number of antinodes formed in $W_1$ to $W_2$ is

In an experiment to study standing waves,you use a string whose mass per unit length is $\mu = (1.0 \pm 0.1) \times 10^{-4} \ kg/m$. You look at the fundamental mode,whose frequency $f$ is related to the length $L$ and tension $T$ of the string by the equation $L = \frac{1}{2f} \sqrt{\frac{T}{\mu}}$. You make a plot with $L$ on the $y$-axis and $\sqrt{T}$ on the $x$-axis,and find that the best-fitting line is $y = (8.0 \pm 0.3) \times 10^{-3}x + (0.2 \pm 0.04)$ in $SI$ units. What is the value of the frequency of the wave (including the error)? Express your result in $SI$ units $(Hz)$.

Two uniform stretched strings $A$ and $B$,made of steel,are vibrating under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$,the ratio of the lengths of the strings is