Question

A source of frequency \(f\) gives 5 beats/sec when sounded with a source of frequency 200 Hz. The second harmonic of \(f\) gives 10 beats/sec when sounded with a source of frequency 420 Hz. The value of \(f\) is:

• A. 195 Hz
• B. 205 Hz
• C. 190 Hz
• D. 210 Hz
• E.

Hint:

When you have multiple choices for the frequency from the first set of beats, always calculate the harmonic for each and see which one produces the required beat frequency with the second source. It acts as an automatic verification.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
Beat frequency is the absolute difference between the frequencies of two sounding sources: \(f_{beat} = |f_1 - f_2|\). Harmonic frequencies are integer multiples of the fundamental frequency (\(2^{nd}\) harmonic = \(2f\)).

Step 2: Key Formula or Approach
1. \(|f - 200| = 5\).
2. \(|2f - 420| = 10\).

Step 3: Detailed Explanation
1. From the first condition, \(f\) can be either: - \(f = 200 + 5 = 205\) Hz - \(f = 200 - 5 = 195\) Hz
2. Now, test both possibilities with the second condition (\(2^{nd}\) harmonic produces 10 beats with 420 Hz): - If \(f = 205\) Hz: \(2f = 2 \times 205 = 410\) Hz. Beat frequency with 420 Hz = \(|420 - 410| = 10\) Hz. (Matches given condition) - If \(f = 195\) Hz: \(2f = 2 \times 195 = 390\) Hz. Beat frequency with 420 Hz = \(|420 - 390| = 30\) Hz. (Does not match)

Step 4: Final Answer
The value of \(f\) is 205 Hz.