If ${n_1},{n_2},{n_3}.........$ are the frequencies of segments of a stretched string, the frequency $n$ of the string is given by

A block $\mathrm{M}$ hangs vertically at the bottom end of a uniform rope of constant mass per unit length. The top end of the rope is attached to a fixed rigid support at $O$. A transverse wave pulse (Pulse $1$ ) of wavelength $\lambda_0$ is produced at point $O$ on the rope. The pulse takes time $T_{O A}$ to reach point $A$. If the wave pulse of wavelength $\lambda_0$ is produced at point $A$ (Pulse $2$) without disturbing the position of $M$ it takes time $T_{A 0}$ to reach point $O$. Which of the following options is/are correct?

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[$A$] The time $\mathrm{T}_{A 0}=\mathrm{T}_{\mathrm{OA}}$

[$B$] The velocities of the two pulses (Pulse $1$ and Pulse $2$) are the same at the midpoint of rope.

[$C$] The wavelength of Pulse $1$ becomes longer when it reaches point $A$.

[$D$] The velocity of any pulse along the rope is independent of its frequency and wavelength.

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

The fundamental frequency of a sonometer with a weight of $4\,kg$ is $256\,Hz$ . The weight required to produce its octave is .... $kg-wt$

Two identical strings $X$ and $Z$ made of same material have tension $T _{ x }$ and $T _{ z }$ in them. If their fundamental frequencies are $450\, Hz$ and $300\, Hz ,$ respectively, then the ratio $T _{ x } / T _{ z }$ is$.....$

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 waves is set up in the composite wire, the joint is found to be a node. The ratio of the number of an tin odes formed in $W_1$ to $W_2$ is