Q60PP - University Physics with Modern Physics

Question

The professor once again returns the apparatus to its original setting, but now she adjusts the oscillator to produce sound waves of half the original frequency. What happens? (a) The students who originally heard a loud tone again hear a loud tone, and the students who originally heard nothing still hear nothing.(b) The students who originally heard a loud tone now hear nothing, and the students who originally heard nothing now hear a loud tone. (c) Some of the students who originally heard a loud tone again hear a loud tone, but others in that group now hear nothing. (d) Among the students who originally heard nothing, some still hear nothing but others now hear a loud tone.

Step-by-Step Solution

Answer

The correct option is C

Step 1: Important Concepts

Constructive interference occurs at a point where the superposing waves are in phase and destructive interference occurs at points where the two waves are out of phase.

Step 2: Find new wavelength

Since the frequency of sound changes and the speed of sound is constant, the wavelength of sound must change too to keep the speed of sound same

                                             $$\begin{gathered} v_s={\lambda }_{new}f \\ {\lambda }_{new}=\frac{v_s}{f} \end{gathered}$$         

                                                       

Since the new frequency is half the original ie  $$\begin{gathered} f=\frac{f_0}{2} \\ =\frac{500 HZ}{2} \\ =250 Hz \end{gathered}$$

                                              $$\begin{gathered} {\lambda }_{new}=\frac{340 m/s}{250 Hz} \\ =1.36 m \end{gathered}$$

 

The ratio is given by 

                                              

                                     ${\lambda }_{new}=2{\lambda }_0$                     

Step 3: Change in interference location

Earlier when the wavelength was unchanged then the students at the following positons were able to hear the sound

                                     $\Delta x=m{\lambda }_0$                 

Where m is an integer,

Students at  $\Delta x={\lambda }_0,2{\lambda }_0,3{\lambda }_0.....$ would hear the maximum sound.

 

When the wavelength is changed, path difference is given by 

                                       $\Delta x_{new}=2m{\lambda }_0$                      

 

This means that at the position in which sound was initially maximum, the interference in the new case for students at  ${\lambda }_0,3{\lambda }_0,5{\lambda }_0.....$ would be the destructive one 

But the interference in the new case for the students at $2{\lambda }_0,4{\lambda }_0,6{\lambda }_0.....$  will stay constructive.

Step 4: Conclusion

Therefore, some students who initially heard a loud tone will hear again a loud tone, but others win this group will now hear nothing. Hence the correct option is C