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 tuning fork and a sonometer wire were sounded together and produce $4$ beats per second. When the length of sonometer wire is $95 cm$ or $100 cm,$ the frequency of the tuning fork is ….. $Hz$

Standing waves are produced in $10 \,m$ long stretched string fixed at both ends. If the string vibrates in $5$ segments and wave velocity is $20 \,m / s$, the frequency is ……. $Hz$

A sonometer wire resonates with a given tuning fork forming standing waves with five antinodes between the two bridges when a mass of $9\,kg$ is suspended from the wire. When this mass is replaced by a mass $M,$ the wire resonates with the same tuning fork forming three antinodes for the same positions of the bridges. The value of $M$ is …. $kg$

A sonometer wire oflength $1.5\ m$ is made of steel. The tension in it produces an elastic strain of $1 \%$. What is the fundamental frequency of steel if density and elasticity of steel are $7.7 \times 10^3 $ $kg/m^3$ and $2.2 \times 10^{11}$ $N/m^2$ respectively?