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

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?

(image)

[$A$] 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.

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 total length of a sonometer wire between fixed ends is $110\, cm$. Two bridges are placed to divide the length of wire in ratio $6 : 3 : 2$. The tension in the wire is $400\, N$ and the mass per unit length is $0.01\, kg/m$ . What is the minimum common frequency with Which three parts can vibrate ……….. $Hz$ ?

Two open organ pipes of fundamental frequencies $n_{1}$ and $n_{2}$ are joined in series. The fundamental frequecny of the new pipe so obtained will be