AIIMS 2006 Physics Sound and Beats MCQ Question

To solve the problem of determining the frequency of the guitar string when it is sounded with tuning forks of different frequencies, we can use the concept of beat frequency. The beat frequency is the difference in frequencies between two sound sources.

Step-by-Step Explanation:

1. Understanding Beat Frequency: The beat frequency $f_b$ is given by the absolute difference between the frequency of the two sound sources: $f_b = |f_s - f_t|$ where $f_s$ is the frequency of the guitar string and $f_t$ is the frequency of the tuning fork.

2. Using the First Tuning Fork (440 Hz): For the first tuning fork with a frequency of 440 Hz, the beat frequency observed is 5 Hz: $f_b = |f_s - 440| = 5$

   This gives us two equations:

   • $f_s - 440 = 5$ (1)
   • $440 - f_s = 5$ (2)

   From equation (1): $f_s = 440 + 5 = 445 \, \text{Hz}$

   From equation (2): $f_s = 440 - 5 = 435 \, \text{Hz}$

   So, the possible frequencies of the string from this tuning fork are $445 \, \text{Hz}$ or $435 \, \text{Hz}$.

3. Using the Second Tuning Fork (437 Hz): For the second tuning fork with a frequency of 437 Hz, the beat frequency observed is 8 Hz: $f_b = |f_s - 437| = 8$

   This gives us two more equations:

   • $f_s - 437 = 8$ (3)
   • $437 - f_s = 8$ (4)

   From equation (3): $f_s = 437 + 8 = 445 \, \text{Hz}$

   From equation (4): $f_s = 437 - 8 = 429 \, \text{Hz}$

   So, the possible frequencies from this tuning fork are $445 \, \text{Hz}$ or $429 \, \text{Hz}$.

4. Finding the Correct Frequency: Now we combine the results from both tuning forks:

   • From the first tuning fork, we found possible frequencies of $445 \, \text{Hz}$ or $435 \, \text{Hz}$.
   • From the second tuning fork, we found possible frequencies of $445 \, \text{Hz}$ or $429 \, \text{Hz}$.

   The only frequency that appears in both sets of possible frequencies is $445 \, \text{Hz}$.

Conclusion:

Therefore, the frequency of the guitar string is: $f_s = 445 \, \text{Hz}$

Clarifying Other Options:

• Option B (435 Hz): This frequency fits the first tuning fork’s results but does not satisfy the second tuning fork’s results. Thus, it cannot be the correct answer.
• Option C (429 Hz): This frequency fits the second tuning fork’s results but does not satisfy the first tuning fork’s results. It is also not the correct answer.
• Option D (448 Hz): This option does not match any of the derived frequencies and is therefore incorrect.

Thus, the correct answer is A) 445 Hz.