The equation of a standing wave in a string fixed at both ends is given as $y=2 A \sin k x \cos \omega t$ The amplitude and frequency of a particle vibrating at the mid of an antinode and a node are respectively

For a transverse wave travelling along a straight line, the distance between two peaks (crests) is $5 \,m ,$ while the distance between one crest and one trough is $1.5 \,m$ The possible wavelengths (in $m$ ) of the waves are

Two uniform strings of mass per unit length $\mu$ and $4 \mu$, and length $L$ and $2 L$, respectively, are joined at point $O$, and tied at two fixed ends $P$ and $Q$, as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$, which of the following statement($s$) is(are) correct?

$(A)$ With a node at $O$, the minimum frequency of vibration of the composite string is $v_0$

$(B)$ With an antinode at $O$, the minimum frequency of vibration of the composite string is $2 v_0$

$(C)$ When the composite string vibrates at the minimum frequency with a node at $O$, it has $6$ nodes, including the end nodes

$(D)$ No vibrational mode with an antinode at $O$ is possible for the composite string

A piano string $1.5\,m$ long is made of steel of density $7.7 \times 10^3 \,kg/m^3$ and  Young’s modulus $2 \times 10^{11} \,N/m^2$. It is maintained at a tension which produces  an elastic strain of $1\%$ in the string. The fundamental frequency of transverse vibrations of string is ......... $Hz$

Two similar sonometer wires given fundamental frequencies of $500Hz$. These have same tensions. By what amount the tension be increased in one wire so that the two wires produce $5$ beats/sec .... $\%$

The string of a violin has a frequency of $440 \,cps$. If the violin string is shortened by one fifth, its frequency will be changed to ........... $cps$