Beats - UNSOLVED PRACTICE SET

Q7. What are beats? Under what conditions are they produced?

Q8. Two tuning forks produce 4 beats per second. The frequency of one fork is 256 Hz. What are the possible frequencies of the other fork?

Q9. Derive the expression for the beat frequency when two waves of frequencies f₁ and f₂ superpose.

Q10. Why can't we hear beats if the difference in frequencies of the two sources is more than about 10 Hz?

Q11. A tuning fork of unknown frequency produces 5 beats per second with a standard fork of 340 Hz. When a little wax is put on the unknown fork, the beat frequency decreases. What was the original frequency of the unknown fork? Explain your reasoning.

Q12. How is the phenomenon of beats used to tune musical instruments?

Q13. Using the principle of superposition, derive the expression for beats. Consider two waves:

y₁ = A sin(2πf₁t) and y₂ = A sin(2πf₂t)

Show that:

(i) The resultant wave has a frequency equal to the average of f₁ and f₂

(ii) The amplitude varies with time at the beat frequency |f₁ – f₂|

(iii) The intensity varies at twice the beat frequency

Draw a graph showing the variation of resultant displacement with time.

Q14. Discuss the applications of beats in:

(i) Tuning musical instruments

(ii) Determining unknown frequencies

(iii) Detecting dangerous gases in mines (acoustic gas analysis)

(iv) Radar and sonar technology

For each application, explain the underlying principle and how beats provide useful information.

Q15. A student sets up an experiment with two tuning forks mounted on resonance boxes.

(i) Describe how she can demonstrate the phenomenon of beats.

(ii) How can she determine which fork has the higher frequency when beats are heard?

(iii) She loads one fork with wax and notices the beat frequency changes. Explain why.

(iv) Design an experiment to measure the frequency of an unknown tuning fork using a known fork and the phenomenon of beats.

Q16. Two tuning forks A and B produce 6 beats per second. Fork A has a frequency of 340 Hz. Fork B is loaded with a little wax and now produces 4 beats per second with fork A.

(i) What were the possible original frequencies of fork B?

(ii) Which of these is the correct frequency? Justify your answer.

(iii) What is the frequency of fork B after loading with wax?

(iv) If the beat frequency had increased instead of decreased after loading, what would be the frequency of fork B?

(v) Explain why loading a tuning fork with wax decreases its frequency.

Q17. Two sound sources produce waves described by:

y₁ = 0.01 sin(800πt)

y₂ = 0.01 sin(804πt)

(i) Calculate the frequencies of the two sources.

(ii) Calculate the beat frequency.

(iii) Write the expression for the resultant wave.

(iv) Calculate the time interval between two successive maxima of sound.

(v) Calculate the time interval between a maximum and the next minimum of sound.

Q18. In a school music room, a student is tuning a guitar string. She uses a tuning fork of 440 Hz as a reference.

(i) She hears 3 beats per second. What are the possible frequencies of the guitar string?

(ii) She tightens the string slightly and now hears 5 beats per second. What was the original frequency of the string? Explain your reasoning.

(iii) How many turns of the tuning peg are needed to bring the string to exactly 440 Hz if each turn changes the frequency by 2 Hz?

(iv) The student then checks another string that should be at 330 Hz. She hears 2 beats per second with a 330 Hz fork. She loosens the string and the beats disappear. Explain why loosening worked for this string but tightening worked for the previous one.

(v) A piano tuner uses this method to tune all 88 keys. Why is this method more accurate than using an electronic tuner for developing a musician's ear?