Tag: superposition of waves-2: stationary (standing) waves: vibrations of air columns

Questions Related to superposition of waves-2: stationary (standing) waves: vibrations of air columns

Assertion - In a stationary wave, no transfer of energy takes place.
Reason - There is no onward motion of the disturbance from one particle to adjoining particle in stationary wave.

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns from moving to stationary stationary (or standing) waves formation of stationary waves

  1. Assertion and Reason are correct and Reason is the correct explanation for Assertion

  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

  3. Assertion is correct but Reason is incorrect

  4. Both Assertion and Reason are incorrect

Show answer
Correct Option: B
Explanation:

In stationary wave, total energy associated with it is twice the energy of each of incident and reflected wave. large amount of energy are stored equally in standing wave and became trapped with wave. Hence there is no transmission of energy through the waves.

As two students holds opposite ends of slinky that is the resting on the floor, one student shakes the end he is holding back and forth with the constant frequency. He later shakes exactly the same way but with a much greater frequency.
Which statement best describes other changes that take place as a result of this increased frequency?

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns from moving to stationary stationary (or standing) waves formation of stationary waves

  1. The wave speed and the wavelength both increases.

  2. The wave speed increases, but the wavelength does not significantly change.

  3. The wavelength increases ,but the wave speed does not significantly change.

  4. The wave speed decreases, but the wavelength does not significantly change.

  5. The wavelength decreases ,but the wave speed does not significantly change.

Show answer
Correct Option: A
Explanation:

We know that frequency of a wave is characterized by the source of wave . In both the cases wave in the slinky has a constant frequency but not same in both cases  , 

now we have $v=f\lambda$ ,
or                  $f=v/\lambda$ , 
when frequency $f$ is  increased in second case , wave speed $v$ increases and to maintain a constant frequency $f$ (to maintain a constant ratio), wavelength $\lambda$ also increases
 .

In a standing wave on a string.

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns from moving to stationary stationary (or standing) waves formation of stationary waves

  1. In one time period all the particles are simultaneously at rest twice.

  2. All the particles must be at their positive extremes simultaneously once in one time period.

  3. All the particles may be at their positive extremes simultaneously once in a time period.

  4. All the particles are never at rest simultaneously.

Show answer
Correct Option: A,C
Explanation:

Standing waves are obtained when two waves with same angular frequencies and velocity are superimposed, (They are however moving in the opposite directions).
$x(t) = A\sin(\omega t - kx) + A\sin(\omega t + kx + \delta)$
$x(t) = 2A\cos(kx)\sin(\omega t +\dfrac{\delta}{2})$
For all particles to be simultaneously at rest, the value of the sine function must be equal to zero.
i.e. $\omega t + \dfrac{\delta}{2} = n\pi$
$\Rightarrow$ $t = \dfrac{1}{\omega}(n\pi - \dfrac{\delta}{2})$
$\omega = \dfrac{2\pi}{T}$
$\Rightarrow$ $t = \dfrac{T}{2\pi}(n\pi - \dfrac{\delta}{2})$
$t _{1} =  \dfrac{T}{2\pi}(n\pi - \dfrac{\delta}{2})$
$t _{2} =  \dfrac{T}{2\pi}((n+1)\pi - \dfrac{\delta}{2})$
$t _{2} - t _{1} = \dfrac{T}{2}$


So, the time between this event happening twice is half to time period, so in one cycle this would happen twice. So Option (A)
For the particle to be at positive extreme the sine function can take a value of 1 only.
It can be shown that this happens at an interval of '$T$'
So it will happen twice in a time period if the displacement is max at the start of the time period and once more at the end of the time period,
or else it would happen only once in a time period. Hence Option (C)

Equation of a standing wave is expressed as y = 2A sin$\omega$t coskx. In the equation, quantity $\omega$/k represents

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns from moving to stationary stationary (or standing) waves formation of stationary waves

  1. the transverse speed of the particles of the string.

  2. the speed of the component waves

  3. the speed of the standing wave.

  4. a quantity that is independent of the properties of the string.

Show answer
Correct Option: A
Explanation:

$y=1A\sin\omega t\cos kx\k=\cfrac{2\pi}{\lambda}\ \Rightarrow \cfrac{\omega}{K}=\cfrac{\omega\times \lambda}{2\pi}=\cfrac{\lambda}{2\pi/\omega}=\cfrac{\lambda}{T}=\lambda f=V$

$\Rightarrow\cfrac{\omega}{K}=$ Velocity of wave particles (transverse speed)

As an empty vessel is filled with water, its fundamental frequency

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. Increases

  2. Decreases

  3. Remains the same

  4. None of these

Show answer
Correct Option: A
Explanation:

   An empty vessel with a base is like a closed organ pipe , and fundamental frequency of a closed organ pipe is given by ,

       $n _{1}=v/4l$ ,
where $l=$ length of air column in pipe (height of pipe) ,
  from above we get,    
       $n _{1}\propto 1/l$ ,
when the empty vessel is filled with water , the length of air column $l$ in pipe decreases and therefore fundamental frequency $n _{1}$ increases as length of air column and fundamental frequency are inversely proportional to each other .

A resonance tube apparatus is employed to.

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. Investigate the dependence of velocity of sound in air upon temperature

  2. Verify the laws of vibrating strings

  3. Study beats

  4. Determine the velocity of sound in air

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Correct Option: D

In a resonating air column, the first booming sound is heard when the length of air column is $10\ cm$. The second booming sound will be heard when length is:

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. $20\ cm$

  2. $30\ cm$

  3. $40\ cm$

  4. None of the above

Show answer
Correct Option: B
Explanation:

Booming sound indicates that at that length, $l _1$, the air column is in resonance with the given frequency.
and that length is,
$l _1= \lambda /4=10$
or, $\lambda = 40cm$
The next resonance length will be :
$l _2=3\lambda/4=30 cm$

In a resonace air column experiment, first and second resonance are obtained at length of air columns $l _{1}$ and $l _{2}$ the third resonance will be obtained at a length of

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. $2l _{2}-l _{1}$

  2. $l _{2}-2l _{1}$

  3. $l _{2}-l _{1}$

  4. $3l _{2}-l _{1}$

Show answer
Correct Option: A

If in an experiment for determination of velocity of sound by resonance tube method using a tuning fork of 512 Hz, first resonance was observed at 30.7 cm and second was obtained at 63.2 cm , then maximum possible error in velocity of sound is ( consider actual speed of sound in air is 332 m/s )

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. $204$ $cm/sec$

  2. $280$ $cm/sec$

  3. $58$ $cm/sec$

  4. $80$ $cm/sec$

Show answer
Correct Option: B

Which is(are) the factor(s) on which the frequency of sound emitted due to vibration in an air column depends?

physics superposition of waves-2: stationary (standing) waves: vibrations of air columns determining wavelength and speed of sound resonance tube resonance and sonometer

  1. Length of air column

  2. Diameter of air column

  3. Both A and B

  4. None of these

Show answer
Correct Option: C
Explanation:

In a closed air column, the frequency of mth mode of vibration is given by:

          $f _{m}=\dfrac{v(2m-1)}{4l}$      m=1,2,3......,
In a open air column, the frequency of mth mode of vibration is given by:
          $f _{m}=\dfrac{vm}{2l}$      m=1,2,3......
We can see that frequency depends upon the length of air column $l$ .  
Rayleigh stated that in an air column the antinodes are not formed exactly at the open end, but slightly outside. The distance of the antinode from the antinode is called end correction (e), therefore the length of the air column will change due to end correction hence frequency, this end correction depends upon the radius, r (or diameter) of air column and is equal to $0.6r $.      
Therefore frequency of vibration depends upon on both length and diameter of air column .