Embibe Experts Solutions for Chapter: Mechanical Wave, Exercise 1: JEE Advanced Paper 2 - 2019

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_s$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_0$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_s$, will the sound reaching the pipe lead to a resonance?

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.
 

Answer the following by appropriately matching the lists based on the information given in the paragraph.
A musical instrument is made using four different metal strings, $1,\mathrm{ }2,\mathrm{ }3$ and $4$ with mass per unit length $\mathrm{\mu },2\mathrm{\mu },3\mathrm{\mu }$ and $4\mathrm{\mu }$ respectively. The instrument is played by vibrating the strings by varying the free length in between the range ${\mathrm{L}}_0$ and $2{\mathrm{L}}_0.$ It is found that in string- $1\left(\mathrm{\mu }\right)$ at free length ${\mathrm{L}}_0$ and tension ${\mathrm{T}}_0$ the fundamental mode frequency is ${\mathrm{f}}_0.$
List-I gives the above four strings while list-II lists the magnitude of some quantity.
 

In an experiment, to measure the speed of sound by a resonating air column, a tuning fork of frequency $500 \mathrm{Hz}$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length $50.7\mathrm{cm}$ and $83.9\mathrm{cm}$. Which of the following statements is (are) true?

A stationary tuning fork is in resonance with an air column in a pipe. If the tuning fork is moved with a speed of $2 \mathrm{m} {\mathrm{s}}^{-1}$ in front of the open end of the pipe and parallel to it, the length of the pipe should be changed for the resonance to occur with the moving tuning fork. If the speed of sound in air is $320 \mathrm{m} {\mathrm{s}}^{-1},$ the smallest value of the percentage change required in the length of the pipe is ________.

A train $\mathrm{S}1,$ moving with a uniform velocity of $108\mathrm{ }\mathrm{k}\mathrm{m}/\mathrm{h},$ approaches another train $\mathrm{S}2$ standing on a platform. An observer $\mathrm{O}$ moves with a uniform velocity of $36\mathrm{ }\mathrm{k}\mathrm{m}/\mathrm{h}$ towards $\mathrm{S}2,$ as shown in figure. Both the trains are blowing whistles of same frequency $120\mathrm{ }\mathrm{H}\mathrm{z}.$ When $\mathrm{O}$ is $600\mathrm{ }\mathrm{m}$ away from $\mathrm{S}2$ and distance between $\mathrm{S}1$ and $\mathrm{S}2$ is $800\mathrm{ }\mathrm{m},$ the number of beats heard by $\mathrm{O}$ is____.

[Speed of the sound $=330\mathrm{ }\mathrm{m} {\mathrm{s}}^{-1}]$

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are defined as $\overrightarrow{A}=a\widehat{i}$ and $\overrightarrow{B}=a\left(\cos \omega t\widehat{i}+\sin \omega t\widehat{j}\right)$, where a is a constant and $\omega =\frac{\pi }{6} \mathrm{rad}\mathrm{ }{\mathrm{s}}^{-1}.$ If $\left|\overrightarrow{A}+\overrightarrow{B}\right|=\sqrt{3}\left|\overrightarrow{A}-\overrightarrow{B}\right|$ at time $t=\tau$ for the first time, the value of $\tau$ in seconds, is_______

Two men are walking along a horizontal straight line in the same direction. The man in front walks at a speed $1.0\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{s}}^{-1}$ and the man behind walks at a speed $2.0\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{s}}^{-1}$ . A third man is standing at a height $12\mathrm{ }\mathrm{m}$ above the same horizontal line such that all three men are in a vertical plane. The two walking men are blowing identical whistles which emit a sound of frequency $1430\mathrm{ }\mathrm{Hz}$ . The speed of sound in air is $330\mathrm{ }\mathrm{m}\mathrm{ }{\mathrm{s}}^{-1}$. At the instant, when the moving men are $10\mathrm{ }\mathrm{m}$ apart, the stationary man is equidistant from them. The frequency of beats in $\mathrm{Hz}$ , heard by the stationary man at this instant, is __________.