The transverse displacement in a streched string is given by
$y = 0.06 \sin \, \left( {\frac{{2\pi }}{3}x} \right)\cos \,(120\pi t)$

where $x$ and $y$ are in $m$ and $t$ is in $s$. The length of the string is $1.5\, m$ and  its mass is $3.0 \times 10^{-2} \,kg$, then tension in string is ….. $N$

A uniform string resonates with a tuning fork, at a maximum tension of $32 \,N$. If it is divided into two segments by placing a wedge at a distance one-fourth of length from one end, then to resonance with same frequency the maximum value of tension for string will be ……….. $N$

The length of a sonometer wire is $0.75\, m$ and density $9 \times 10^3\, kg/m^3$. It can bear a stress of $8.1 \times 10^8\, N/m^2$ without exceeding the elastic limit. What is the fundamental frequency that can be produced in the wire  …. $Hz$ ?

If $n _{1}, n_{2}$ and $n _{3}$ are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency $n$ of the string is given by

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 $\mathrm{C}^{\prime}$.